# Consecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$

N. A. Carella

Abstract: Let $p$ be a large prime, and let $k\ll\log p$. A new proof of the
existence of any pattern of $k$ consecutive quadratic residues and quadratic
nonresidues is introduced in this note. Further, an application to the least
quadratic nonresidues $n_{p}$ modulo $p$ shows that $n_{p}\ll(\log p)(\log\log
p)$. ††
MSC2020: Primary 11A15, Secondary 11L40.
Keywords: Least quadratic nonresidue, Consecutive quadratic residues.

##  1 Introduction

Given a prime $p\geq 2$, a nonzero element $u\in\mathbb{F}_{p}$ is called a
quadratic residue, equivalently, a square modulo $p$ whenever the quadratic
congruence $x^{2}-u\equiv 0\operatorname{mod}p$ is solvable. Otherwise, it
called a quadratic nonresidue. A finite field $\mathbb{F}_{p}$ contains
$(p+1)/2$ squares $\mathcal{R}=\\{u^{2}\operatorname{mod}p:0\leq u<p/2\\}$,
including zero. The quadratic residues are uniformly distributed over the
interval $[1,p-1]$. Likewise, the quadratic nonresidues are uniformly
distributed over the same interval.

Let $k\geq 1$ be a small integer. This note is concerned with the longest runs
of consecutive quadratic residues and consecutive quadratic nonresidues (or
any pattern)

$u,\quad u+1,\quad u+2,\quad\ldots,\quad u+k-1,$ (1)

in the finite field $\mathbb{F}_{p}$, and large subsets
$\mathcal{A}\subset\mathbb{F}_{p}$. Let $N(k,p)$ be a tally of the number of
sequences 1.

######  Theorem 1.1.

Let $p\geq 2$ be a large prime, and let $k=O\left(\log p\right)$ be an
integer. Then, the finite field $\mathbb{F}_{p}$ contains $k$ consecutive
quadratic residues (or quadratic nonresidues or any pattern). Furthermore, the
number of $k$ tuples has the asymptotic formulas

  1. (i)

$\displaystyle
N(k,p)=\frac{p}{2^{k}}\left(1-\frac{1}{p}\right)^{k}\left(1+O\left(\frac{1}{p}\right)\right),$
8cm if $k\geq 1.$

  2. (ii)

$\displaystyle N(k,p)=\frac{p}{2^{k}}+O\left(k^{2}\right),$8cm if $k\geq 1.$

The first expression is suitable for applications requiring any integer $k$,
and the second expression is suitable for applications requiring small
integers $k$. The proofs are assembled in Section 8. The main results are
proved using a new counting technique, based on Lemma 5.2. It provides sharper
error terms than the standard technique based on Lemma 5.1.

######  Theorem 1.2.

Let $p\geq 2$ be a large prime, and let $k=O\left(\log p\right)$ be an
integer. Then, for any subset of consecutive elements
$\mathcal{A}\subset\mathbb{F}_{p}$ of cardinality
$p^{1-\varepsilon/2}\ll\\#\mathcal{A}$ contains $k$ consecutive quadratic
residues (or quadratic nonresidues or any pattern), $\varepsilon>0$ is an
arbitrary small number. Furthermore, the number of $k$ tuples has the
asymptotic formulas

  1. (i)

$\displaystyle
N(k,p,\mathcal{A})=\frac{\\#\mathcal{A}}{2^{k}}\left(1-\frac{1}{p}\right)^{k}\left(1+O\left(\frac{1}{p}\right)\right),$
8cm if $k\geq 1.$

  2. (ii)

$\displaystyle
N(k,p,\mathcal{A})=\frac{\\#\mathcal{A}}{2^{k}}+O\left(k\right),$8cm if $k\geq
1.$

Quadratic residues $r\in\mathbb{F}_{p}$ (and quadratic non residues) in finite
fields have orders $\operatorname{ord}_{p}(r)=2$. The analysis and results for
$k$ consecutive $d$ power residues or any pattern of $d$ power residues have
similar details, but are more complex as the orders of the elements increases.
The other result consider an interesting application to the least quadratic
nonresidue.

The other result considers an interesting application to the least quadratic
nonresidue $n_{p}$ modulo $p$. The current unconditional result for the least
quadratic nonresidues in the literature states that

$n_{p}\ll p^{1/4\sqrt{e}+\varepsilon},$ (2)

where $\varepsilon>0$, and the strongest conditional result for primitive
character $\chi$ states that

$n_{\chi}(p)\ll(\log p)^{1.37+o(1)},$ (3)

see [4, Corollary 2], and Conjecture 10.1. The following result is proved
here.

######  Theorem 1.3.

For any large prime $p\geq 2$, the least quadratic nonresidue is bounded by

$n_{p}\ll(\log p)(\log\log p).$ (4)

The implied constant should be small, perhaps $\leq 20$, see Table 1 in
Section 11. In Section 10 several Lemmas are spliced together to prove this
result. Section 3 to Section LABEL:s5000 cover the supporting materials and
other optional topics.

##  2 Quadratic Symbol

######  Definition 2.1.

Let $p\geq 2$ be a prime, and let $u\in\mathbb{F}_{p}$. The quadratic symbol
modulo $p$ of a nonzero element $u$ is defined by

$\left(\frac{u}{p}\right)=\begin{cases}1&\text{ if }u\text{ is a quadratic
residue};\\\ -1&\text{ if }u\text{ is not a quadratic residue}.\\\
\end{cases}$ (5)

In term of calculations, this can be determined via the Euler criterion

$\left(\frac{u}{p}\right)=u^{(p-1)/2}\equiv\pm 1\operatorname{mod}p.$ (6)

######  Lemma 2.1.

(Legendre) Let $p\geq 2$ and $q$ be a pair of distinct primes. Then, the
quadratic symbol satisfies the following properties.

  1. (i)

$\displaystyle\left(\frac{a}{p}\right)\equiv a^{(p-1)/2}\operatorname{mod}p,$
8cm Euler congruence equation.

  2. (ii)

$\displaystyle\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right),$8cm
completely multiplicative function.

  3. (iii)

$\displaystyle\left(\frac{-1}{p}\right)=\left(-1\right)^{(p-1)/2},$8cm
evaluation at $a=-1$.

  4. (iv)

$\displaystyle\left(\frac{2}{p}\right)=\left(-1\right)^{(p^{2}-1)/8},$8cm
evaluation at $a=2$.

  5. (v)

$\displaystyle\left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=\left(-1\right)^{(p-1)(q-1)/4},$8cm
quadratic reciprocity.

##  3 Some Quadratic Exponential Sums

######  Definition 3.1.

Let $p\geq 2$ be a prime. The finite Fourier transform of a periodic function
$f:\mathbb{Z}\longrightarrow\mathbb{C}$ of period $p$ is defined by

$\hat{f}(s)=\frac{1}{\eta_{p}\sqrt{p}}\sum_{t\in\mathbb{F}_{p}}f(t)e^{i2\pi
st/p},$ (7)

where $\eta_{p}=1$ if $p\equiv 1\operatorname{mod}4$ or $\eta_{p}=i$ if
$p\equiv 3\operatorname{mod}4$.

Except for a normalizing factor, the standard Gauss sum is a finite Fourier
transform of the nonprincipal character
$\chi:\mathbb{Z}\longrightarrow\mathbb{C}$, namely,

$\tau_{s}(\chi)=\sum_{t\in\mathbb{F}_{p}}\chi(t)e^{i2\pi ts/p}.$ (8)

######  Lemma 3.1.

The quadratic character mod $p$ is the unique fix point of the finite Fourier
transform. Specifically

$\left(\frac{s}{p}\right)=\frac{1}{\eta_{p}\sqrt{p}}\sum_{t\in\mathbb{F}_{p}}\left(\frac{t}{p}\right)e^{i2\pi
st/p},$ (9)

where $\eta_{p}=1$ if $p\equiv 1\operatorname{mod}4$ or $\eta_{p}=i$ if
$p\equiv 3\operatorname{mod}4$.

######  Lemma 3.2.

(Gauss) If $p\geq 2$ is a prime, then

$\sum_{u\in\mathbb{F}_{p}}e^{i2\pi u^{2}/p}=\begin{cases}\sqrt{p}\text{ if
}p\equiv 1\operatorname{mod}p;\\\ i\sqrt{p}\text{ if }p\equiv
3\operatorname{mod}p.\\\ \end{cases}$ (10)

###### Proof.

It is widely available in the literature, exampli gratia, [7, Theorem 1.1.5],
[15, Section 3.3], [20, Lemma 3.3]. ∎

######  Lemma 3.3.

Let $p\geq 2$ be a prime, and let $(x\,|\,p)$ be the quadratic character mod
$p$. If the element $s\neq 0$, then,

$\sum_{u\in\mathbb{F}_{p}}e^{i2\pi
u^{2}s/p}=\left(\frac{s^{-1}}{p}\right)\eta_{p}\sqrt{p},$ (11)

where $\eta_{p}=1$ if $p\equiv 1\operatorname{mod}4$ or $\eta_{p}=i$ if
$p\equiv 3\operatorname{mod}4$.

###### Proof.

Let $\chi(n)=(x\,|\,p)$. Replace the characteristic function of quadratic
residue, see Lemma 5.1, to obtain

$\sum_{u\in\mathbb{F}_{p}}e^{i2\pi
u^{2}s/p}=\sum_{u\in\mathbb{F}_{p}}\left(1+\chi(u)\right)e^{i2\pi
us/p}=\sum_{u\in\mathbb{F}_{p}}\chi(u)e^{i2\pi us/p}.$ (12)

The change of variable $z=us$ returns

$\sum_{u\in\mathbb{F}_{p}}e^{i2\pi
u^{2}s/p}=\chi(s^{-1})\sum_{z\in\mathbb{F}_{p}}\chi(z)e^{i2\pi
z/p}=\chi(s^{-1})\eta_{p}\sqrt{p}.$ (13)

∎

######  Lemma 3.4.

If $p\geq 2$ is a prime, and $a\neq$ is an integer such that $\gcd(a,p)=1$,
then

$\sum_{u\in\mathbb{F}_{p}}\left(\frac{ax^{2}+bx+c}{p}\right)=\begin{cases}-\left(\frac{a}{p}\right)&\text{
if }b^{2}-4ac\not\equiv 0\operatorname{mod}p;\\\
\left(\frac{a}{p}\right)(p-1)&\text{ if }b^{2}-4ac\equiv
0\operatorname{mod}p.\\\ \end{cases}$ (14)

###### Proof.

Consult the literature, [7, Theorem 2.1.2], [17], and similar references. ∎

##  4 Some Incomplete Exponential Sums

A classical application of the finite Fourier transform provides nontrivial
upper bounds of incomplete character sums. The simplest one uses the quadratic
symbol or plain character $\chi$ modulo $q$.

######  Lemma 4.1.

(Polya-Vinogradov) For $q$ is a large prime, and character $\chi$ modulo $q$,

$\sum_{n\leq x}\chi(n)\ll\sqrt{q}\log q.$ (15)

###### Proof.

Use the finite Fourier transform of $\chi(n)$ as in Lemma 3.1 and the
geometric series, and other means. ∎

The distribution, and various properties of the implied constant has a vast
literature, and it is a topic of current research, see [5], [12], et alii.
Many improved upper bounds for some specific characters such as $\chi(-1)=-1$
or $\chi(-1)=1$ are known. An explicit for the Burgess inequality is stated
below.

######  Lemma 4.2.

([30]) Let $p\geq 10^{7}$ be a prime, and let $\chi$ be character modulo $p$.
Let $M$, and $N\geq 1$ be nonnegative integer and let $r\geq 1$. Then

$\sum_{M\leq n\leq N+M}\chi(n)\leq 2.7N^{1-1/r}p^{(r+1)/4r^{2}}(\log
p)^{1/r}.$ (16)

At $r=1$ it reduces to the Polya-Vinogradov inequality, and as $r\to\infty$,
it becomes a trivial upper bound.

######  Lemma 4.3.

Suppose that GRH is true. Then, for any nonprincipal character $\chi$ modulo
$q$ and any large number $x$,

$\sum_{n\leq x}\chi(n)\ll\sqrt{q}\log\log q.$ (17)

###### Proof.

This is done in [22, Theorem 2]. ∎

######  Lemma 4.4.

(Paley) There are infinitely many discriminant $q\equiv 1\operatorname{mod}q$
for which

$\sum_{n\leq x}\left(\frac{n}{q}\right)\gg\sqrt{q}\log\log q.$ (18)

###### Proof.

The original version appears in [26] and a recent version is given in [21,
Theorem 9.24]. ∎

##  5 Characteristic Functions For Quadratic Residues

The standard characteristic function of quadratic residues and quadratic
nonresidues are induced by the quadratic symbol.

######  Lemma 5.1.

Let $p\geq 2$ be a prime, and let $(x\,|\,p)$ be the quadratic character mod
$p$. If $u\in\mathbb{F}_{p}$ is a nonzero element, then,

  1. (i)

$\displaystyle\Psi_{2}\left(u\right)=\frac{1}{2}\left(1+\left(\frac{u}{p}\right)\right)=\left\\{\begin{array}[]{ll}1&\text{
if }u^{(p-1)/2}\equiv 1\operatorname{mod}p,\\\ 0&\text{ if
}u^{(p-1)/2}\equiv-1\operatorname{mod}p.\\\ \end{array}\right.$

  2. (ii)

$\displaystyle\overline{\Psi}_{2}(u)=\frac{1}{2}\left(1-\left(\frac{u}{p}\right)\right)=\left\\{\begin{array}[]{ll}1&\text{
if }u^{(p-1)/2}\equiv-1\operatorname{mod}p,\\\ 0&\text{ if }u^{(p-1)/2}\equiv
1\operatorname{mod}p,\\\ \end{array}\right.$

are the characteristic functions for quadratic residues and quadratic non
residues modulo $p$ respectively in the finite field $\mathbb{F}_{p}$.

A new representation of the characteristic function for quadratic residues and
quadratic nonresidues are introduced below.

######  Lemma 5.2.

Let $p\geq 2$ be a prime, and let $\tau$ be a primitive root mod $p$. If
$u\in\mathbb{F}_{p}$ is a nonzero element, then,

  1. (i)

$\displaystyle\Psi_{2}(u)=\sum_{0\leq n<(p-1)/2}\frac{1}{p}\sum_{0\leq m\leq
p-1}e^{i2\pi\left(\tau^{2n}-u\right)m/p}=\left\\{\begin{array}[]{ll}1&\text{
if }u^{(p-1)/2}\equiv 1\operatorname{mod}p,\\\ 0&\text{ if
}u^{(p-1)/2}\equiv-1\operatorname{mod}p.\\\ \end{array}\right.$

  2. (ii)

$\displaystyle\overline{\Psi}_{2}(u)=\sum_{0\leq
n<(p-1)/2}\frac{1}{p}\sum_{0\leq m\leq
p-1}e^{i2\pi\left(\tau^{2n+1}-u\right)m/p}=\left\\{\begin{array}[]{ll}1&\text{
if }u^{(p-1)/2}\equiv-1\operatorname{mod}p,\\\ 0&\text{ if }u^{(p-1)/2}\equiv
1\operatorname{mod}p.\\\ \end{array}\right.$

###### Proof.

(i) For a fixed $u\neq 0$, the finite field $\mathbb{F}_{p}$ equation

$\tau^{2n}-u=0$ (19)

has a unique solution $n=n_{0}\in\\{0,1,2,\ldots(p-1)/2-1\\}$ if and only if
$0\neq u=\tau^{2n_{0}}$ is a quadratic residue modulo $p$. This, in turns,
implies that the inner exponential sum collapses to $p$. Specifically,

$\sum_{0\leq m\leq
p-1}e^{i2\pi\left(\tau^{2n}-u\right)m/p}=\left\\{\begin{array}[]{ll}p&\text{
if }u^{(p-1)/2}\equiv 1\operatorname{mod}p,\\\ 0&\text{ if
}u^{(p-1)/2}\equiv-1\operatorname{mod}p.\\\ \end{array}\right.$ (20)

Otherwise, $\tau^{2n}-u\neq 0$, which implies that the inner exponential sum
vanishes. ∎

##  6 Estimate For The Sum $T(k,p)$

The calculations for the sum

$T(k,p)=\sum_{0\neq n\in\mathbb{F}_{p}}\prod_{0\leq i\leq
k-1}\left(\frac{1}{p}\sum_{0\leq n_{i}<(p-1)/2}1\right)$ (21)

assumes the existence of a sequence of $k$ consecutive quadratic residues

$u,\quad u+1,\quad u+2,\quad\ldots,\quad u+k-1$ (22)

in the finite field $\mathbb{F}_{p}$. However, it is valid for any pattern of
quadratic residues and nonresidues.

######  Lemma 6.1.

Let $p\geq 2$ be a large prime, let $k=O\left(\log p\right)$ be an integer,
then,

  1. (i)

$\displaystyle T(k,p)=\frac{p-1}{2^{k}}\left(1-\frac{1}{p}\right)^{k},$ 8cm if
$k\geq 1.$

  2. (ii)

$\displaystyle T(k,p)=\frac{p}{2^{k}}+O\left(k\right),$8cm if $k\geq 1.$

###### Proof.

(i) Routine calculations return

$\displaystyle T(k,p)$ $\displaystyle=$ $\displaystyle\sum_{0\neq
n\in\mathbb{F}_{p}}\prod_{0\leq i\leq k-1}\left(\frac{1}{p}\sum_{0\leq
n_{i}<(p-1)/2}1\right)$ $\displaystyle=$ $\displaystyle\sum_{0\neq
n\in\mathbb{F}_{p}}\left(\frac{1}{p}\cdot\frac{p-1}{2}\right)^{k}$
$\displaystyle=$
$\displaystyle\frac{p-1}{2^{k}}\left(1-\frac{1}{p}\right)^{k}.$

(ii) For small integer $k=O\left(\log p\right)$, the binomial series leads to

$\displaystyle T(k,p)$ $\displaystyle=$
$\displaystyle\frac{p-1}{2^{k}}\left(1-\frac{1}{p}\right)^{k}$
$\displaystyle=$
$\displaystyle\frac{p-1}{2^{k}}\left(1-\binom{k}{1}\frac{1}{p}+\binom{k}{2}\frac{1}{p^{2}}+\cdots+\frac{(-1)^{k}}{p^{k}}\right)$
$\displaystyle\leq$
$\displaystyle\frac{p-1}{2^{k}}\left(1+k\binom{k}{k/2}\frac{1}{p}\right)$
$\displaystyle=$ $\displaystyle\frac{p}{2^{k}}+O\left(k\right),$

since the central binomial coefficient $\binom{k}{k/2}\leq 2^{k}$. ∎

##  7 The Estimates For The Sum $U(k,p)$

The exponential sums over finite fields $\mathbb{F}_{p}$ studied in Section 3
are used to estimate the sum

$U(k,p)=\sum_{0\neq u\in\mathbb{F}_{p}}\prod_{0\leq r\leq
k-1}\left(\frac{1}{p}\sum_{\begin{subarray}{c}0\leq n_{r}<(p-1)/2\\\
0<m_{r}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{r}}-u-a_{r})m_{r}\right)}\right),$
(25)

which is used to prove Theorem 1.1.

######  Lemma 7.1.

Let $p\geq 2$ be a large prime, let $k\geq 1$ be an integer, and let $\tau$ be
a primitive root mod $p$. If the elements $u+a_{r}=v_{r}^{2}\neq 0$ are
quadratic residues for $r=0,1,2,...,k-1$, then,

  1. (i)

$\displaystyle
U(k,p)=O\left(\frac{1}{2^{k}}\left(1+\frac{1}{p}\right)^{k}\right),$ 8cm if
$k\geq 1.$

  2. (ii)

$\displaystyle U(k,p)=O\left(\frac{k}{2^{k}}\right),$8cm if $k\geq 1.$

###### Proof.

(i) Rewrite the multiple finite sum (25) as a product

$\displaystyle U(k,p)$ $\displaystyle=$ $\displaystyle\sum_{0\neq
u\in\mathbb{F}_{p}}\left(\frac{1}{p}\sum_{\begin{subarray}{c}0\leq
n_{0}<(p-1)/2\\\ 0<m_{0}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{0}}-u-a_{0})m_{0}\right)}\right)$
$\displaystyle\qquad\qquad\times\prod_{1\leq r\leq
k-1}\left(\frac{1}{p}\sum_{\begin{subarray}{c}0\leq n_{r}<(p-1)/2\\\
0<m_{r}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{r}}-u-a_{r})m_{r}\right)}\right)$
$\displaystyle=$ $\displaystyle\sum_{0\neq u\in\mathbb{F}_{p}}U_{1}\times
U_{2}.$

Merging Lemma 7.2 and Lemma 7.3, (using Holder inequality is optional), return

$\displaystyle U(k,p)$ $\displaystyle=$ $\displaystyle\sum_{0\neq
u\in\mathbb{F}_{p}}U_{1}\cdot U_{2}$ $\displaystyle=$
$\displaystyle\sum_{0\neq u\in\mathbb{F}_{p}}\frac{e^{-i2\pi
a_{0}/p}}{4p}\left(1+\eta_{p}^{2}p\left(\frac{u^{-1}}{p}\right)\right)\cdot\left(\frac{p+1}{2p}\right)^{k-1}$
$\displaystyle=$ $\displaystyle\frac{e^{-i2\pi
a_{0}/p}}{4p}\left(\frac{p+1}{2p}\right)^{k-1}\sum_{0\neq
u\in\mathbb{F}_{p}}1+\frac{\eta_{p}^{2}e^{-i2\pi
a_{0}/p}}{4}\left(\frac{p+1}{2p}\right)^{k-1}\sum_{0\neq
u\in\mathbb{F}_{p}}\left(\frac{u^{-1}}{p}\right)$ $\displaystyle=$
$\displaystyle\frac{e^{-i2\pi
a_{0}/p}}{4p}\left(\frac{p+1}{2p}\right)^{k-1}(p-1),$

since

$\sum_{0\neq u\in\mathbb{F}_{p}}\left(\frac{u^{-1}}{p}\right)=0.$ (28)

The absolute value $\left|U(k,p)\right|$ of the last expression in (7) can be
rewritten as in the Lemma. (ii) Same as the proof in Lemma 6.1-ii. ∎

######  Lemma 7.2.

Let $p\geq 2$ be a large prime, and let $\tau$ be a primitive root mod $p$. If
the element $u+a_{0}=v_{0}^{2}\neq 0$ is a quadratic residue, then,

$U_{1}=\frac{1}{p}\sum_{\begin{subarray}{c}0\leq n<(p-1)/2\\\ 0<m\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n}-u-a_{0})m\right)}=\frac{e^{-i2\pi
a_{0}/p}}{4p}\left(1+\eta_{p}^{2}p\left(\frac{u^{-1}}{p}\right)\right),$ (29)

where $|\eta_{p}|=\pm 1$.

###### Proof.

Rearrange the finite sum as

$\displaystyle U_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{p}\sum_{\begin{subarray}{c}0\leq n<(p-1)/2\\\ 0<m\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n}-u-a_{0})m\right)}$ (30)
$\displaystyle=$ $\displaystyle\frac{e^{-i2\pi a_{0}/p}}{p}\sum_{0<m\leq
p-1}e^{-i2\pi um/p}\sum_{0\leq n<(p-1)/2}e^{i2\pi m\tau^{2n}/p}$
$\displaystyle=$ $\displaystyle\frac{e^{-i2\pi a_{0}/p}}{2p}\sum_{0<m\leq
p-1}e^{-i2\pi um/p}\sum_{0\leq n<p-1}e^{i2\pi m\tau^{2n}/p}.$

Let $s=\tau^{n}$. An application of Lemma 3.2, yields

$1+2\sum_{0\leq n<p-1}e^{i2\pi m\tau^{2n}/p}=\sum_{0\leq n<p-1}e^{i2\pi
ms^{2}/p}=\left(\frac{m^{-1}}{p}\right)\eta_{p}\sqrt{p}.$ (31)

Substituting this into the previous equation returns

$\displaystyle U_{1}$ $\displaystyle=$ $\displaystyle\frac{e^{-i2\pi
a_{0}/p}}{4p}\sum_{0<m\leq p-1}e^{-i2\pi
um/p}\left(-1+\left(\frac{m^{-1}}{p}\right)\eta_{p}\sqrt{p}\right)$
$\displaystyle=$ $\displaystyle\frac{e^{-i2\pi
a_{0}/p}}{4p}\left(1+\eta_{p}\sqrt{p}\sum_{0<m\leq
p-1}\left(\frac{m^{-1}}{p}\right)e^{-i2\pi um/p}\right)$ $\displaystyle=$
$\displaystyle\frac{e^{-i2\pi
a_{0}/p}}{4p}\left(1+\eta_{p}^{2}p\left(\frac{u^{-1}}{p}\right)\right),$

where $|\eta_{p}|=\pm 1$. ∎

######  Lemma 7.3.

Let $p\geq 2$ be a large prime, and let $\tau$ be a primitive root mod $p$. If
the elements $u+a_{r}=v_{r}^{2}\neq 0$ are quadratic nonresidues, then,

$U_{2}=\prod_{1\leq r\leq k-1}\left(\frac{1}{p}\sum_{\begin{subarray}{c}0\leq
n_{r}<(p-1)/2\\\ 0<m_{r}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{r}}-u-a_{r})m_{r}\right)}\right)=\left(\frac{p+1}{2p}\right)^{k-1}.$
(33)

###### Proof.

The hypothesis $u+a_{r}=v_{r}^{2}$ for $r=1,2,\ldots,k-1$, is used to
determine the value of each incomplete exponential sum. Start with a complete
exponential sum and break it up into two subsums:

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\frac{1}{p}\sum_{\begin{subarray}{c}0\leq n_{r}<(p-1)/2\\\ 0\leq
m_{r}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{r}}-u-a_{r})m_{r}\right)}$
$\displaystyle=$ $\displaystyle\frac{1}{p}\sum_{0\leq
n_{r}<(p-1)/2}1\quad+\quad\frac{1}{p}\sum_{\begin{subarray}{c}0\leq
n_{r}<(p-1)/2\\\ 0<m_{r}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{r}}-u-a_{r})m_{r}\right)}$
$\displaystyle=$
$\displaystyle\frac{p-1}{2p}\quad+\quad\frac{1}{p}\sum_{\begin{subarray}{c}0\leq
n_{r}<(p-1)/2\\\ 0<m_{r}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{r}}-u-a_{r})m_{r}\right)}.$

Solving for the incomplete exponential sum on the right side of (7) yields

$\frac{p+1}{2p}=1-\frac{p-1}{2p},$ (35)

which is basically the trivial value. Taking the product of all the incomplete
exponential sums yields

$\displaystyle U_{2}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{p}\sum_{\begin{subarray}{c}0\leq n_{1}<(p-1)/2\\\
0<m_{1}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{1}}-u-a_{1})m_{1}\right)}\right)$
(36)
$\displaystyle\qquad\qquad\times\cdots\times\left(\frac{1}{p}\sum_{\begin{subarray}{c}0\leq
n_{1}<(p-1)/2\\\ 0<m_{1}\leq
p-1\end{subarray}}e^{i2\pi\left((\tau^{2n_{k-1}}-u-a_{k-1})m_{k-1}\right)}\right)$
$\displaystyle=$ $\displaystyle\frac{p+1}{2p}\times\cdots\times\frac{p+1}{2p}$
$\displaystyle=$ $\displaystyle\left(\frac{p+1}{2p}\right)^{k-1}.$

∎

##  8 Consecutive Quadratic Residues And Nonresidues

Consecutive quadratic nonresidues is one of the simplest configuration of a
subset of two or more consecutive quadratic nonresidues. The earliest attempts
are surveyed in [13], [14], et alii. A more general result was proved by
Carlitz [10, Theorem 3] using a counting technique based on Lemma 5.1. More
precisely, the number of $k$ consecutive quadratic residue symbols or any
pattern of quadratic residue and quadratic nonresidue symbols

$\left(\frac{u}{p}\right)=\varepsilon_{0},\quad\left(\frac{u+1}{p}\right)=\epsilon_{1},\quad\cdots,\quad\left(\frac{u+k-1}{p}\right)=\epsilon_{k-1},$
(37)

where $\epsilon_{n}=\pm 1$, in the finite field $\mathbb{F}_{p}$ has the
asymptotic formula

$\displaystyle N(k,p)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{k}}\sum_{0\leq u\leq
p-1}\left(1+\left(\frac{u}{p}\right)\epsilon_{0}\right)\left(1+\left(\frac{u+1}{p}\right)\epsilon_{1}\right)\cdots\left(1+\left(\frac{u+k-1}{p}\right)\epsilon_{k-1}\right)$
(38) $\displaystyle=$ $\displaystyle\frac{p}{2^{k}}+E(k,p),$

see Lemma 5.1 for details on the the characteristic functions. An explicit
error term $E(k,p)=\pm(k+1)(3+\sqrt{p})$ is proved in [25, Corollary 5]. A
slightly different proof appears in a new survey [20, Theorem 5.6]. For $k\leq
2$, the exponential sums involved have exact evaluations, and there are no
error terms. But, in general, for $k\geq 3$, with very few exceptions, the
exponential sums are estimated, and have error terms of the forms
$E(k,p)=O(k\sqrt{p})$. A new and sharper proof and counting technique based on
Lemma 5.2 is given here.

Let $a_{0},a_{1},a_{2},\ldots,a_{k-1}$ be a sequence of distinct and
increasing integers. Let $p\geq 2$ be a large prime, and let
$\tau\in\mathbb{F}_{p}$ be a primitive root. A pattern of $k$ consecutive
quadratic residues and quadratic nonresidues
$u+a_{0},u+a_{1},u+a_{2},\ldots,u+a_{k-1}$ exists if and only if the system of
equations

$\tau^{2n_{0}}=u+a_{0},\quad\tau^{2n_{1}}=u+a_{1},\quad\tau^{2n_{2}}=u+a_{2},\quad\ldots,\quad\tau^{2n_{k-1}}=u+a_{k-1},$
(39)

has one or more solutions. A solution consists of a $k$-tuple
$n_{0},n_{1},\ldots,n_{k-1}$ of integers such that $0\leq n_{i}<(p-1)/2$ for
$i=0,1,\ldots,k-1$, and some $u\in\mathbb{F}_{p}$. Let

$N(k,p)=\\#\left\\{u\in\mathbb{F}_{p}:\operatorname{ord}_{p}(u+a_{i})=2\right\\}$
(40)

for $i=0,1,\ldots,k-1$, denotes the number of solutions.

###### Proof.

(Theorem 1.1): The total number of solutions is written in terms of
characteristic function for quadratic residues, see Lemma 5.2, as

$\displaystyle N(k,p)$ $\displaystyle=$ $\displaystyle\sum_{0\neq
u\in\mathbb{F}_{p}}\Psi_{2}\left(u+a_{0}\right)\Psi_{2}\left(u+a_{1}\right)\cdots\Psi_{2}\left(u+a_{k-1}\right)$
$\displaystyle=$ $\displaystyle\sum_{0\neq u\in\mathbb{F}_{p}}\prod_{0\leq
i\leq k-1}\left(\frac{1}{p}\sum_{\begin{subarray}{c}0\leq n_{i}\leq(p-1)/2\\\
0\leq m_{i}\leq
p-1\end{subarray}}\psi\left((\tau^{2n_{i}}-u-a_{i})m_{i})\right)\right)$
$\displaystyle=$ $\displaystyle T(k,p)\quad+\quad U(k,p).$

The term $T(k,p)$, which is determined by the indices
$m_{0}=m_{1}=\cdots=m_{k-1}=0$, has the form

$T(k,p)=\sum_{0\neq u\in\mathbb{F}_{p}}\prod_{0\leq i\leq
k-1}\left(\frac{1}{p}\sum_{0\leq n_{i}<(p-1)/2}1\right),$ (42)

and the term $U(k,p)$, which is determined by the indices $m_{0}\neq
0,m_{1}\neq 0,\ldots,m_{k-1}\neq 0$, has the form

$U(k,p)=\sum_{0\neq u\in\mathbb{F}_{p}}\prod_{0\leq i\leq
k-1}\left(\frac{1}{p}\sum_{\begin{subarray}{c}0\leq n_{i}\leq(p-1)/2\\\ 1\leq
m_{i}\leq
p-1\end{subarray}}\psi\left((\tau^{2n_{i}}-u-a_{i})m_{i})\right)\right).$ (43)

(i) Applying Lemma 6.1-i to the term $T(k,p)$, and Lemma 7.1-i to the term
$U(k,p)$, yield

$\displaystyle N(k,p)$ $\displaystyle=$ $\displaystyle T(k,p)\quad+\quad
U(k,p)$ $\displaystyle=$
$\displaystyle\frac{p-1}{2^{k}}\left(1-\frac{1}{p}\right)^{k}+O\left(\frac{1}{2^{k}}\left(1+\frac{1}{p}\right)^{k}\right)$
$\displaystyle=$
$\displaystyle\frac{p}{2^{k}}\left(1-\frac{1}{p}\right)^{k}\;\left(1+O\left(\frac{1}{p}\right)\right)$
$\displaystyle>$ $\displaystyle 0,$

for all sufficiently large primes $p\geq 2$.

(ii) Applying Lemma 6.1-ii to the term $T(k,p)$, and Lemma 7.1 to the term
$U(k,p)$, yield

$\displaystyle N(k,p)$ $\displaystyle=$ $\displaystyle T(k,p)\quad+\quad
U(k,p)$ $\displaystyle=$
$\displaystyle\frac{p}{2^{k}}+O\left(k\right)+O\left(\frac{k}{2^{k}}\right)$
$\displaystyle=$ $\displaystyle\frac{p}{2^{k}}+O\left(k\right)$
$\displaystyle>$ $\displaystyle 0,$

for all sufficiently large primes $p\geq 2$. ∎

##  9 Synopsis Of Upper Bounds For Quadratic Nonresidues

The mathematical literature has many estimates for the least quadratic
nonresidue $n_{p}$ modulo $p$. A list of the most frequently encountered upper
bounds is complied below.

  1. (1)

$\displaystyle n_{p}\leq p^{1/2}+1$, 4cmderived using an ad hoc elementary
argument, see [23, Theorem 3.9 ], [29], etc.

  2. (2)

$\displaystyle n_{p}\ll p^{1/2}\log p$, 4cmderived from the Polya-Vinogradov
inequality.

  3. (3)

$\displaystyle n_{p}\leq p^{1/2\sqrt{e}+\varepsilon}$, 4cmderived from the
Polya-Vinogradov inequality for any $\varepsilon>0$.

  4. (4)

$\displaystyle n_{p}\leq p^{1/4\sqrt{e}+\varepsilon}$, 4cmderived from the
Burgess inequality for any $\varepsilon>0$, see Lemma 4.2, and [2].

  5. (5)

$\displaystyle n_{p}\ll p^{\varepsilon}$, 4cmthe Vinogradov conjecture, where
$\varepsilon>0$, see [16], and the literature.

  6. (6)

$\displaystyle n_{p}\leq 2(\log p)^{2}$, 4cmderived from the GRH, see [1],
[3].

##  10 The Least Quadratic Nonresidue

Two slightly different heuristics for the conjectured upper bound of the least
quadratic nonresidue are given in [24] and [30]. These are summarized below.

######  Conjecture 10.1.

For every large prime $p\geq 3$,

$n_{p}\ll(\log p)(\log\log p).$ (46)

The heuristic is based on the proportion of primes $p$ such that the $n$th
prime $p_{n}$ is the least quadratic nonresidue modulo $p$. The proportion has
a geometric distribution, and its proof is based on quadratic reciprocity and
Dirichlet theorem for primes in arithmetic progressions. For example, the
probability for each $n\geq 1$ is given by limit

$P(n_{p}=p_{n})=\lim_{x\to\infty}\frac{\\#\\{p\leq
x:n_{p}=p_{n}\\}}{\pi(x)}=\frac{1}{2^{n}}.$ (47)

Note that the form of the main term in Theorem 1.1-ii implies that quadratic
residues (or quadratic non residues) in a finite field $\mathbb{F}_{p}$ are
independent or nearly independent random variables $X=X(p)$ with probability

$P\left(\operatorname{ord}_{p}\left(X\right)=2\right)=\frac{1}{2}+O\left(\frac{1}{p^{\varepsilon}}\right),$
(48)

where $\varepsilon>0$ is an arbitrary small number.

On average, the expected value $\overline{n}$ of the least quadratic residue
$n_{p}$ is quite small

$\overline{n}=\lim_{x\to\infty}\frac{1}{\pi(x)}\sum_{2<p\leq
x}n_{p}=\sum_{n\geq 1}\frac{p_{n}}{2^{n}}=3.67464\ldots,$ (49)

where $\pi(x)=\\{p\leq x\\}$, and $p_{n}$ is the $n$th prime, see [6, p. 253],
and [19] for the generalized concept.

###### Proof.

(Theorem 1.3) To obtain a reductio ad absurdum, suppose that there exists a
prime $p\geq 2$ for which $n_{p}>(\log p)(\log\log p)$. Let $k=(\log
p)(\log\log p)$. This implies that the finite field $\mathbb{F}_{p}$ contains
a sequence of $k$ consecutive quadratic residues

$u,\quad u+1,\quad u+2,\quad\ldots,\quad u+k-1.$ (50)

This immediately implies that

$\left(\frac{u}{p}\right)=1,\quad\left(\frac{u+1}{p}\right)=1,\quad\cdots,\quad\left(\frac{u+k-1}{p}\right)=1.$
(51)

By Theorem 1.1, the total number of such sequences of quadratic residues of
length $k$ is

$N(k,p)=\frac{p}{2^{k}}\left(1-\frac{1}{p}\right)^{k}\left(1+O\left(\frac{1}{p}\right)\right)\gg
1.$ (52)

Taking logarithm, and simplifying return

$\displaystyle\log p-k\log
2+k\log\left(1-\frac{1}{p}\right)+\log\left(1+O\left(\frac{1}{p}\right)\right)\gg
0.$ (53)

Rearranging it, and replacing $k=(\log p)(\log\log p)$ give

$\displaystyle\log p$ $\displaystyle\gg$ $\displaystyle k\log
2+O\left(\frac{k}{p}\right)-\log\left(1+O\left(\frac{1}{p}\right)\right)$
$\displaystyle\gg$ $\displaystyle(\log p)(\log\log p)\log 2+O\left(\frac{(\log
p)(\log\log p)}{p}\right)+\log C_{0},$

where $C_{0}\approx 1$. Clearly, this is false. Hence, a finite field
$\mathbb{F}_{p}$ contains a quadratic nonresidues $n_{p}\ll(\log p)(\log\log
p)$. ∎

The best upper bound for the parameter $H\geq 0$ for which a character
$\chi(n)$ modulo $p$ is constant on the interval $[N,N+H]$ is
$H<7.07p^{1/4}\log p$ for large primes, see [18, Theorem 1.1], and [31]. The
above result in Theorem 1.3 provides an improved and effective upper bound
$H\ll(\log p)(\log\log p)$ for the parameter $H\geq 0$. While the result
$N(k,p)=p/2^{k}+O(k\sqrt{p})$ in [25, Corollary 5] provides an effective lower
bound $H\gg\log p$.

######  Corollary 10.1.

Let $p\geq 2$ be a large prime, let $\chi$ be a nonprincipal character modulo
$p$, and let $x\geq 1$ be a large real number. Define the real value function

$f(x)=\sum_{0\leq n\leq x}\chi(n).$ (55)

Then, $f:\mathbb{R}\longrightarrow\mathbb{Z}$ satisfies the following
properties.

  1. (i)

$\displaystyle f(x)=f(x+p)$, 7cmis periodic of period $p$.

  2. (ii)

$\displaystyle f(x)\ll p^{1/2}\log p$, 7cmis of absolute bounded variation on
the real line.

  3. (iii)

$\displaystyle f(N)<f(N+1)<\cdots<f(N+H)$, 7cmis monotonically increasing on a
short interval $[N,N+H]$ if and only if $H\ll(\log p)(\log\log p)$, for any
$N\geq 0$.

  4. (iv)

$\displaystyle f(N)>f(N+1)>\cdots>f(N+H)$, 7cmis monotonically decreasing on a
short interval $[N,N+H]$ if and only if $H\ll(\log p)(\log\log p)$, for any
$N\geq 0$.

##  11 Experimental Data

The numerical data [20, Table 1], an expanded version is duplicated below,
suggests that the constant is $c_{p}=n_{p}/(\log p)(\log\log p)\leq 20$ for
all primes $p\geq 2$.

Table 1: Numerical Data for the Least Quadratic Nonresidue Modulo $p$. $n$ | $n_{p}=p_{n}$ | $p$ | $(\log p)(\log\log p)$ | $c_{p}$
---|---|---|---|---
1 | $2$ | $3$ | $0.10$ | $20.00$
2 | $3$ | $7$ | $1.30$ | $2.31$
3 | $5$ | $23$ | $3.58$ | $1.40$
4 | $7$ | $71$ | $6.18$ | $1.13$
5 | $11$ | $311$ | $10.03$ | $1.10$
6 | $13$ | $479$ | $11.23$ | $1.16$
7 | $17$ | $1559$ | $14.67$ | $1.16$
8 | $19$ | $5711$ | $18.66$ | $1.02$
9 | $23$ | $10559$ | $20.63$ | $1.11$
10 | $29$ | $18191$ | $22.40$ | $1.29$
11 | $31$ | $31391$ | $24.20$ | $1.28$
12 | $37$ | $422231$ | $33.18$ | $1.22$
13 | $41$ | $701399$ | $40.00$ | $1.03$
14 | $43$ | $366791$ | $32.68$ | $1.32$
15 | $47$ | $3818929$ | $41.20$ | $1.14$

##  12 Computational Complexity Of Quadratic Residues And Square Roots

The identification of an element $u\in\mathbb{F}_{p}$ as a quadratic residue
(or quadratic nonresidue) has nearly linear deterministic time complexity
$O\left(\log p)(\log\log p)^{c}\right)$ for some $c>0$. The discovery of an
algorithm of linear complexity is an open problem, see [6, p. 3] for details.

The determination of the roots of congruence $x^{2}-u\equiv
0\operatorname{mod}p$ has slightly higher time complexity depending on the
following data.

  1. (1)

The residue class of the prime $p\geq 3$.

  2. (2)

The method used to compute a quadratic nonresidue: probabilistic,
deterministic.

  3. (3)

The method used to compute the roots: probabilistic, deterministic.

The worst case has deterministic time complexity $O\left(\log p)^{4}\right)$
bit operations. Many algorithms such as Cipolla algorithm, tonelli algorithm,
Berlekamp algorithm etc, are explained in [6, p. 157], [11, p. 102], [9]. Some
polynomials for computing the square roots are provided in [8].

##  13 Twin Quadratic Residues And Nonresidues

The precise numbers of pairs $QQ$, $QN$, $NQ$ or $NN$ of quadratic residues
and quadratic nonresidues are proved in [7, Theorem 6.3.1]. The proof based on
Lemma 5.2 is given below.

######  Theorem 13.1.

Let $p\geq 3$ and $a\in\mathbb{Z}$, $\gcd(a,p)=1$. The number of pairs $n$ and
$n+a$ such that $(n|p)=\epsilon_{0}$ and $(n|p)=\epsilon_{1}$ is exactly

$N(\epsilon_{0},\epsilon_{1},P)=\frac{1}{4}\left(p-2-\epsilon_{0}\left(\frac{a}{p}\right)-\epsilon_{1}\left(\frac{-a}{p}\right)-\epsilon_{0}\epsilon_{1}\right),$
(56)

where $\epsilon_{i}=\pm 1$.

###### Proof.

Sum the product of the two characteristic functions over the finite field
$\mathbb{F}_{p}$:

$\displaystyle N(\epsilon_{0},\epsilon_{1},p)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{\begin{subarray}{c}n\in\mathbb{F}_{p}\\\ n\neq
0,n\neq-a\end{subarray}}\left(1+\left(\frac{n}{p}\right)\epsilon_{0}\right)\left(1+\left(\frac{n+a}{p}\right)\epsilon_{1}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(p-2+\epsilon_{0}\sum_{\begin{subarray}{c}n\in\mathbb{F}_{p}\\\
n\neq
0,n\neq-a\end{subarray}}\left(\frac{n}{p}\right)+\epsilon_{1}\sum_{\begin{subarray}{c}n\in\mathbb{F}_{p}\\\
n\neq
0,n\neq-a\end{subarray}}\left(\frac{n+a}{p}\right)+\epsilon_{0}\epsilon_{1}\sum_{\begin{subarray}{c}n\in\mathbb{F}_{p}\\\
n\neq 0,n\neq-a\end{subarray}}\left(\frac{n^{2}+an}{p}\right)\right).$

Use Lemma 3.4 to evaluate

$\sum_{\begin{subarray}{c}n\in\mathbb{F}_{p}\\\ n\neq
0,n\neq-a\end{subarray}}\left(\frac{n^{2}+an}{p}\right)=-1$ (58)

and simplify the expression. ∎

A new proof based on Lemma 5.2 yields the same result up to a small error
term. In particular, at $k=2$, Theorem 1.1 reduces to

$N(\epsilon_{0},\epsilon_{1},P)=N(2,p)=\frac{p}{2^{k}}+O\left(k\right)=\frac{p}{4}+O\left(1\right).$
(59)

##  14 Problems

###  14.1 Square Roots Problems

######  Exercise 14.1.

Let $n\geq 2$ be a squarefree integer, let $\mathcal{U}=\\{u\neq 0:u\cdot
u^{-1}\equiv 1\mod n\\}$ be the subset of units (invertible elements), and let
$\mathcal{Q}=\\{m^{2}\operatorname{mod}n:m\geq 0\\}$ be the subset of squares
in the finite ring $\mathbb{Z}/n\mathbb{Z}$.

  1. (a)

Show that total number of units is $\\#\mathcal{U}=\varphi(n)=n\prod_{p\mid
n}(1-1/p)$ units, where $\varphi(n)$ is the number integers relatively prime
number to $n$.

  2. (b)

Show that total number of squares is $\\#\mathcal{Q}=\prod_{p\mid n}(p+1)/2$
squares modulo $n$.

  3. (c)

Show that a square element $s=r^{2}\in\mathbb{Z}/n\mathbb{Z}$ has
$2^{\omega(n)}$ square roots modulo $n$, where $\omega(n)$ is the number of
prime divisors of $n$. For example, $\sqrt{s}=r_{0},r_{1},\ldots,r_{m-1}$.

######  Exercise 14.2.

Let $n\geq 2$ be an integer, and let
$\mathcal{Q_{0}}=\\{m^{2}\operatorname{mod}n:m\geq\\}-\\{0\\}$ be the subset
of squares. Show that the subset of squares
$\\#\mathcal{Q}\cup\mathbb{Z}/n\mathbb{Z}$ is a multiplicative subgroup.

######  Exercise 14.3.

Let $n\geq 2$ be an integer, and let
$\mathcal{Q}=\\{m^{2}\operatorname{mod}n:m\geq 1\\}$ be the subset of squares.
Find a formula for the total number of the subset of squares $\\#\mathcal{Q}$.

######  Exercise 14.4.

Let $n=pq\geq 6$, where $p\geq 2$ and $q\geq 2$ are distinct primes, and let
$m=\varphi(n)$ the number of units in the finite ring
$\mathbb{Z}/n\mathbb{Z}$, and let
$\mathcal{Q}=\\{m^{2}\operatorname{mod}n:m\geq 1\\}$ be the subset of squares.
Verify these questions:

  1. (a)

A square $s=r^{2}\in\mathbb{Z}/n\mathbb{Z}$ has $4=2^{\omega(n)}$ square roots
$r_{0},r_{1},r_{2},r_{3}$.

  2. (b)

If $p\equiv q\equiv 3\operatorname{mod}4$, then a single square root
$r_{i}\in\mathcal{Q}$ for some $i=0,1,2,3$.

######  Exercise 14.5.

Let $n=pq\geq 6$, where $p\geq 2$ and $q\geq 2$ are distinct primes, and let
$m=\varphi(n)$ the number of units in the finite ring
$\mathbb{Z}/n\mathbb{Z}$, and let
$\mathcal{Q}=\\{k^{2}\operatorname{mod}n:k\geq 1\\}$ be the subset of squares.
Verify these questions:

  1. (a)

A square $s=r^{2}\in\mathbb{Z}/n\mathbb{Z}$ has $4=2^{\omega(n)}$ square roots
$r_{0},r_{1},r_{2},r_{3}$.

  2. (b)

If $p\equiv q\equiv 1\operatorname{mod}4$, then square roots,
$r_{i}\not\in\mathcal{Q}$ for $i=0,1,2,3$.

######  Exercise 14.6.

Let $n\geq 2$ be an integer, and let
$\mathcal{Q}=\\{m^{2}\operatorname{mod}n:m\geq 1\\}$ be the subset of squares.
Reference: Handbook of Cryptography.

  1. (a)

If $n=pq$, where $p$ and $q$ are primes, and $s\in\mathcal{Q}$, then the
inverse $s^{-1}\equiv s^{((p-1)(q-1)+1)/8}\operatorname{mod}n$.

  2. (b)

If $n=pqr$, , where $p$, $q$, and $r$ are primes, and $s\in\mathcal{Q}$, find
a similar formula for then the inverse $s^{-1}\equiv
s^{?((p-1)(q-1)(r-1)+1)/16}\operatorname{mod}n$.

###  14.2 Character Sums Problems

######  Exercise 14.7.

Let $q\geq 2$ be an integer, let $\chi$ be a multiplicative nonprincipal
character modulo $q$, and $L(s,\chi)=\sum_{n\geq 1}\chi(n)n^{-s}$. Show that

$\sum_{n\leq
x}\chi(n)=\frac{1}{i2\pi}\int_{c-i\infty}^{c+i\infty}L(s,\chi)\frac{x^{s}}{s}ds,$

where $c>1$ is a constant, and $x\in\mathbb{R}-\mathbb{Z}$ is a real number.

###  14.3 Sums Of Squares Problems

######  Exercise 14.8.

Let $p\equiv 1\operatorname{mod}4$ be a prime, and let
$\mathcal{Q}=\\{k^{2}\operatorname{mod}p:k\geq 1\\}$ be the subset of squares.
Show that

$\sum_{r\in\mathcal{Q}}r=\frac{p(p-1)}{4}.$

######  Exercise 14.9.

Let $n\geq 1$ be an integer, and let
$\mathcal{Q}=\\{k^{2}\operatorname{mod}n:k\geq 1\\}$ be the subset of squares.
Classify and evaluate the sums of squares

$\sum_{r\in\mathcal{Q}}r?.$

###  14.4 Distribution And Spacing Between Squares Problems

######  Exercise 14.10.

Use the quadratic reciprocity and Dirichlet theorem for primes in arithmetic
progressions to prove that the proportion of primes $p$ such that the $n$th
prime $p_{n}$ is the least quadratic nonresidue modulo $p$ has a geometric
distribution. For each $n\geq 1$, the probability is given by limit

$P(n_{p}=p_{n})=\lim_{x\to\infty}\frac{\\#\\{p\leq
x:n_{p}=p_{n}\\}}{\pi(x)}=\frac{1}{2^{n}}.$

######  Exercise 14.11.

Use the geometric distribution for the proportion of primes $p$ such that the
$n$th prime $p_{n}$ is the least quadratic nonresidue modulo $p$ to compute
the average least quadratic nonresidue

$\overline{n}=\lim_{x\to\infty}\frac{1}{\pi(x)}\sum_{2<p\leq
x}n_{p}=3.67464\ldots.$

######  Exercise 14.12.

Let $n\geq 2$ be a squarefree integer, and let
$\mathcal{Q}=\\{m^{2}\operatorname{mod}n:m\geq 1\\}$ be the subset of squares.
Show that the average spacing between squares

$\overline{S_{n}}=\frac{n}{\\#\mathcal{Q}}=\frac{n}{\prod_{p\mid
n}(p+1)/2}=\frac{n2^{\omega(n)}}{\psi(n)},$

where $\omega(n)$ is the prime divisors counting function, and
$\psi(n)/n=\prod_{p\mid n}(1+1/p)$ is the Dedekind psi function, see Exercise
14.1.

######  Exercise 14.13.

Let $n\geq 2$ be an integer, and let
$\mathcal{Q}=\\{m^{2}\operatorname{mod}n:m\geq 1\\}$ be the subset of squares.
Find an expression for the average spacing between squares

$\overline{S_{n}}=\frac{n}{\\#\mathcal{Q}},$

in terms of the $\omega(n)$ is the prime divisors counting function, and
$\sigma(n)/n=\prod_{p^{v}\mid\mid n}(1+1/p+\cdots 1p^{v})$ is the sum of
divisors function, see Exercise 14.3.

###  14.5 Algorithm Problems

######  Exercise 14.14.

Let $m,n\in\mathbb{Z}$ be a pair of distinct integers. Construct a
deterministic algorithm to compute a simultaneous quadratic nonresidue $\eta$
modulo both $n$ and $m$. Hint: Try a pair of distinct prime $p$ and $q$ first,
then generalize it.

######  Exercise 14.15.

Determine the time complexity of computing the inverse $s^{-1}\equiv
a\operatorname{mod}p$ using the Euclidean algorithm. Hint: Consider Lame
theorem.

######  Exercise 14.16.

Determine the time complexity of computing the inverse $s^{-1}\equiv
a\operatorname{mod}p$ using the Fermat theorem $s^{p-2}\operatorname{mod}p$.
Hint: Consider the add-multiply algorithm.

###  14.6 Primitive Roots Quadratic Nonresidues Problems

######  Exercise 14.17.

Given a prime $p\geq 3$, prove the followings.

  1. (a)

Show that a primitive root in a finite field $\mathbb{F}_{p}$ must be a
quadratic nonresidue, but a quadratic nonresidue must not be primitive root.

  2. (b)

A finite field $\mathbb{F}_{p}$ has $(p-1)/2-\varphi(p-1)$ quadratic
nonresidues which are not primitive roots.

  3. (c)

Verify that every quadratic nonresidues in the finite field $\mathbb{F}_{p}$
is primitive root if and only if $p=2^{2^{n}}+1$ is prime.

###  14.7 Open Problems

######  Exercise 14.18.

Given a large prime $p\geq 3$, let
$\mathcal{Q}=\\{n^{2}\operatorname{mod}p:n\geq 1\\}$ be the subset of squares,
and let $\mathcal{A},\mathcal{B}\subset\mathbb{F}_{p}$ be a pair of nonempty
subsets. The subsets are proper subsets and must have zero densities in
$\mathbb{F}_{p}$ to avoid trivial cases. Reference: [28], [32].

  1. 1.

(Sarkozy conjecture) Prove or disprove the existence of an additive partition
$\mathcal{Q}=\mathcal{A}+\mathcal{B}$.

  2. 2.

Prove or disprove that every square is a sum of two squares: the existence of
an additive partition $\mathcal{Q}=\mathcal{A}+\mathcal{B}$, where
$\mathcal{A},\mathcal{B}\subset\mathcal{Q}$ are a pair of nonempty proper
subsets.

  3. 3.

Prove or disprove the existence of a difference partition
$\mathcal{Q}=\mathcal{A}-\mathcal{B}$.

######  Exercise 14.19.

(Lehmer conjecture) Given a large prime $p\geq 3$, and let $(x\,|\,p)$ be the
quadratic symbol, and let $a=r^{2}$ and $b=s^{2}$ be a pair of distinct
squares. Prove or disprove the claim that

$\sum_{n\leq
p}\left(\frac{n+a}{p}\right)\left(\frac{n}{p}\right)\left(\frac{n+b}{p}\right)>1$

exists for a finite number of primes $p\geq 2$. Reference: [13, p. 246].

## References

  * [1] Ankeny, N. C. The least quadratic non residue. Ann. of Math. (2) 55 (1952), 65-72.
  * [2] Burgess, D. A. A note on the distribution of residues and nonresidues. J. London Math. Soc. 38 (1963) 253-256.
  * [3] Bach, E. Analytic methods in the analysis and design of number theoretic algorithms, ACM Distinguished Dissertations, MIT Press, MA, 1985.
  * [4] Bober, J. Goldmakher, L. Polya-Vinogradov and the least quadratic nonresidue. arxiv:1311.7556.
  * [5] Bober, J. Goldmakher, L. Granville, A. Koukoulopoulos, D. The frequency and the structure of large character sums. J. Eur. Math. soc. 20 (2018), no. 7, 1759-1818.
  * [6] Bach, Eric; Shallit, Jeffrey. Algorithmic number theory. Vol. 1. Efficient algorithms. Foundations of Computing Series. MIT Press, Cambridge, MA, 1996.
  * [7] Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S. Gauss and Jacobi sums. Canadian Math. Soc. Series of Monographs. A Wiley-Interscience Publication. New York, 1998.
  * [8] N. A. Carella. Formulas for the Square Roots Mod p. arxiv.org/ftp/arxiv/papers/1101/1101.4605.
  * [9] N. A. Carella. Quadratic Nonresidues And Applications. Technical Report 190, Pace University, 2003.
  * [10] Carlitz, L. Sets of primitive roots. Compositio Math. 13 (1956), 65-70.
  * [11] Crandall, Richard; Pomerance, Carl. Prime numbers. A computational perspective. Second edition. Springer, New York, 2005.
  * [12] Fromm, E. Goldmakher, L. Improving the Burgess bound via Polya-Vonogradov. Proc. Amer. Math. Soc. 147 (2019), no. 2, 461-466.
  * [13] Guy, Richard K. Unsolved problems in number theory. Third edition. Problem Books in Mathematics. Springer-Verlag, New York, 2004.
  * [14] Hummel, Patrick. On consecutive quadratic non-residues: a conjecture of Issai Schur. J. Number Theory 103 (2003), no. 2, 257-266. Arxiv:0305298.
  * [15] Lemmermyer, F. Reciprocity Laws. From Euler to Eisenstein. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000.
  * [16] Linnik, Yu. A remark on the least quadratic nonresidue. C. R. (Doklady) Acad Sci. URSS (N.S) 36 (1942), 119-120.
  * [17] Lidl, Rudolf; Niederreiter, Harald. Finite fields. Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
  * [18] McGown, K. On the constant in Burgess bound for the number of consecutive residues or non-residues. Arxiv.1011.4490.
  * [19] G. Martin and P. Pollack. The average least character non-residue and further variations on a theme of Erdos, J. London Math. Soc. 87 (2013) 22-42. Arxiv.1112.1175.
  * [20] Kevin McGown, Enrique Trevino. The least quadratic non-residue. Preprint, July 18, 2019.
  * [21] Montgomery, Hugh L.; Vaughan, Robert C. Multiplicative number theory. I. Classical theory. Cambridge University Press, Cambridge, 2007.
  * [22] Montgomery, Hugh L.; Vaughan, Robert C.  Exponetial sums with multiplicative coefficients. Invnt. Math. 43 (1977), no. 1, 69-82.
  * [23] Niven, I. Zuckerman,H. Montgomerry, H. An Introduction to the Theory of Numbers. Wiley & Sons, New York, 1991.
  * [24] Paul Pollack. The smallest quadratic nonresidue modulo a prime, Slides, August 29, 2012.
  * [25] Peralta, Rene. On the distribution of quadratic residues and nonresidues modulo a prime number. Math. Comp. 58 (1992), no. 197, 433-440.
  * [26] Paley, R. A theorem on Characters, J. London math. soc. 7 (1932), no. 1, 28-32.
  * [27] J.B. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962) 64-94.
  * [28] Igor E. Shparlinski. Additive Decompositions Of Subgroups Of Finite Fields, arxiv.org/pdf/1301.2872.
  * [29] Shapiro, Harold N. Introduction to the theory of numbers. Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley and Sons, Inc., New York, 1983.
  * [30] Trevino, Enrique. The least quadratic non-residue and related problems. Slides, April 17, 2015.
  * [31] Trevino, Enrique. On the maximum number of consecutive integers on which the a character is constant. Mosc. J. Comb. Number Theory 2 (2012), no. 1, 56-81.
  * [32] Vsevolod F. Lev, Jack Sonn. Quadratic residues and difference sets, arXiv:1502.06833.

.


# David Maurice Brink
20 July 1930 - 8 March 2021
Elected FRS 1981

C. V<EMAIL_ADDRESS>A. Bonaccorso2
<EMAIL_ADDRESS>
1Wadham College, Oxford OX1 3PN, U.K, 2INFN, Sezione di Pisa, 56127 Pisa,
Italy.

###### Abstract

David Brink was one of the leading theoretical nuclear physicists of his
generation. He made major contributions to the study of all aspects of nuclear
physics embracing nuclear structure, nuclear scattering, and nuclear
instability. His wide ranging interests and interactions with theorists and
experimentalists alike helped him in providing both theoretical analysis and
interpretations and suggesting experiments. He had the gift of visualising
complex problems in simple terms and provided clear analysis of the underlying
processes. He was an expert on the use of semi-classical methods which
provided an intuitively clear picture of complex phenomena. His research work
and books are characterised by scientific clarity, transparency, and depth.
David possessed outstanding skills in mathematical computation, and he was an
expert on special functions, group theory, and the Feynman path integral
method. David had many research students and collaborated with a large number
of scientists from across the world, for whom he was a source of scientific
and human inspiration and admiration. His most fundamental belief was that
research was a means of trying to discover and understand the beauties of
Nature and explain them in simple terms to others. His absolute belief in the
value of truth and his unselfish and generous attitude in sharing knowledge
makes him an outstanding figure in contemporary Nuclear Physics.

Figure 1: David M. Brink

##  1  Early years and family memories

David Maurice Brink was born on 20 July 1930 in Hobart, Tasmania. His father,
Maurice Brink had been born in the village of Bjuv in Sweden in 1900. David’s
grandparents emigrated to Australia in July 1900. At the age of 14 David’s
father moved to Sydney where he trained to become an accountant. After this he
went to Tasmania and joined an accountancy firm Wise, Lord and Ferguson, where
he eventually became a partner. In 1929 he married Victoria Finlayson, (born
in 1900). Her father David had emigrated with his parents from Scotland in
1884. They had an engineering firm in Devonport, Tasmania whose main activity
was maintaining and repairing machinery for mining, shipping, and timber
companies. David’s grandfather and his colleagues built the first steam car in
Tasmania and between 1900 and 1904 built nine vehicles including three
passenger cars and one 12-passenger bus. David visited his grandparents often
during vacations. He saw the casting floor and other parts of the factory and
enjoyed playing amongst the remains of old steam traction engines.

David was the eldest of three brothers. The Brink brothers went to a Quaker
school in Hobart, Australia 1936 to 1948. David attended the University of
Tasmania during 1948-51 studying Physics, Mathematics, and Chemistry,
graduating with a BSc in December 1950 and was elected as a Rhodes Scholar at
Magdalen College, Oxford, from October 1951. From February 1951 to September
1951 he studied for BSc Honours in Hobart but did not complete the course
because he moved to Oxford in September 1951.

As a student at the University of Tasmania David joined the Hobart Walking
Club. With this club he went on many trips to the interior of the island. When
he arrived in Oxford he became a member of the Oxford University Alpine Club.
Its activities took him to the Alps where he climbed in the Valais and the
Engadine in Switzerland. It was in Switzerland that he met his future wife
Verena. Verena and David married in 1958 and had three children together. His
love for walking was transmitted to his three children who continue to enjoy
walking in urban, rural, and mountainous settings. While always very committed
and absorbed with his Physics he was also a devoted husband and father,
transmitting his joy for walking and travel to his family. He often helped his
children with their homework and was very patient with them, even when they
were not! Together David and his family travelled to, and lived in many
countries across the world, where their horizons were broadened and they were
introduced to the idea that there are many different ways of living and being.
When his children had left home and travelled to other countries he would
often be found in front of an atlas studying their exact whereabouts.

David was very open minded and curious, always accepting other people’s
opinions and points of view. David and Verena were very close, shared
everything and had full respect for each other. Verena was a wonderful host
and the Brinks often organised tea and dinner parties for students, visitors,
and their families. Verena also helped visitors find accommodation, and with
other issues related to living in Oxford. They were also very generous in
offering accommodation at their place whenever possible.

In Oxford David developed an interest in birds, initially just birds he saw in
Oxford, but when he travelled he always liked to look for birds and made lists
of species he saw. This curiosity in nature extended to other species as well,
including trees. When in 1993 he moved to Trento, Italy, he became a member of
the SOSAT, a branch of the alpine club, and went regularly with them on Sunday
trekking trips.

Figure 2: David (right) in Tasmania 1950.

##  2 Graduate studies and Oxford beginnings

David started his studies at Oxford in October 1951. When he arrived at
Magdalen College there was no tutor in Theoretical Physics at the college. His
maths tutor was David Kendal who sent him for tutorials to Jack De Wet at
Balliol College. Jack asked David to read Von Neumann’s book on the
foundations of Quantum Mechanics in German. He also encouraged David to change
his studies from a BA in Mathematics to a D. Phil in Theoretical Physics.
Maurice H. L. Pryce (FRS 1951) was the Wykeham Professor and head of the
Theoretical Physics Department in Oxford from 1946 to 1954. He was David’s
supervisor. Pryce was also the part-time leader of the Theoretical Physics
Division of the Atomic Energy Research Establishment (AERE) at Harwell, not
far from Oxford, where nuclear theory was very much in the forefront and
Rudolf E. Peierls (FRS 1945) was a consultant. At Harwell there was a very
productive theory group including Tony Skyrme and J. P. (Phil) Elliott (FRS
1980). Skyrme organized regular informal meetings known as ’Skyrmishes’.
Important papers in the latest journals were presented and discussed. Members
of the group attended Oxford seminars and while the local group including
Roger Blin-Stoyle (FRS 1976), David Brink, and Pryce attended the Harwell
meetings. Elliott gave some lectures at Oxford on Racah algebra. Later on his
best-known work brought together the shell and collective models to explain
rotational bands in deformed nuclei using the unitary group SU(3). During this
time he wrote a long article in Handbuch der Physik with A. M.(Tony)Lane (FRS
1975) [1] on the shell model.

The foundations of David Brink’s lifelong research, can all be found in his
thesis ”Some Aspects of the Interactions of Fields with Matter” [2] which was
submitted in May 1955. It is a remarkable document for its breadth and early
contributions to the field of nuclear physics. M. Pryce, his thesis adviser
was interested largely in atomic spectroscopy but also studied the
spectroscopy of nuclear energy levels. The advent of the shell model around
1950 opened the door to new theoretical approaches for understanding the
properties of nuclei and applying quantum mechanical tools to calculate them.
There was also a great interest in reactions involving heavy nuclei and which
could only be treated by statistical methods that had been developed much
earlier. Brink’s two-part thesis contained contributions to both areas,
reflecting the interactions between the Harwell and Oxford groups. The first
part was inspired by the shell model and the second contains important
contributions to the statistical theory of nuclear reactions.

In the first part of his thesis, dealing with Nuclear Structure, David
analyzed the spectroscopic consequences of the nucleon-nucleon interaction
acting on the valence nucleons in nuclei close to the doubly-magic 208Pb.
David was able to estimate the order of magnitude of the interaction matrix
elements from the properties of the deuteron. He also proposed treating the
interaction through a density matrix expansion. This would figure prominently
in later work in the field.

The second part of his thesis dealt with reactions involving heavy nuclei. It
was probably inspired by the work of experimental group at Harwell. There, a
Van de Graaff accelerator was used to measure energy levels, moments and
transition rates in nuclei. David was also fortunate to have contact with the
strong experimental group working on neutron resonances. While David was
working on gamma widths of neutron resonances he benefited from contacts with
Prof. Hughes [3] and Prof. Weisskopf who were visiting Oxford. Weisskopf was
very much interested in applying the detailed balance theory to nuclear
reaction and interactions with him must have influenced David because at the
end of the thesis he acknowledges discussions with Victor Weisskopf. The first
subject in this part was the theory of inelastic scattering on deformed
nuclei. David constructed a theory for the excitation of rotational bands in
deformed nuclei based on two new ideas, namely Bohr’s model of deformed nuclei
and the optical model of Weisskopf et al. [4] published the previous year.
David was able to carry out the calculations to a point where the relative
importance of this mechanism in the total cross section could be estimated.
This was an impressive achievement at a time before computers were available
to carry out the full calculations.

The final section of his thesis deals with the decays of the compound-nucleus
resonances produced in reactions on heavy nuclei. The formulas he presented
here are still in use for modeling the spectra and reactions in heavy nuclei
[5]. The best known is the formula for gamma decay rates in compound-nucleus
resonances. This formula is based on a treatment widely known as the ”Brink-
Axel” hypothesis. At a fundamental level, the theory was derived from the
principle of detailed balance which Weisskopf had used very successfully in
other contexts. The principle gives a formula to relate decay rates to
absorption cross sections in the inverse reaction. The Brink-Axel hypothesis
simply states that the absorption cross sections for gamma radiation on
excited states of heavy nuclei can be estimated by the corresponding cross
sections on the ground states. Axel and Brink worked independently. Peter
Axel’s paper appeared in 1962 [6]. The important statement is made on page 101
of David’s thesis and is expressed in equation (11) of Axel’s paper. The
prediction of the statistics of the widths of nuclear resonances, based on the
generalization of the central limit theorem which David had learned about in
his statistics course in Tasmania. David published the results in [7] where he
showed the close connection between the shell-model description of the giant
dipole resonance and the collective model of Goldhaber and Teller [8] and
Steinwedel and Jensen [9]. After his paper, theory of the giant resonances
used the shell model as a starting point. Confirmation of the Brink-Axel
hypothesis first came from the Berkeley experiments in 1981 [10].

The last part of thesis has formulas related to another important topic in
compound-nucleus theory, the fluctuations in decay widths of individual
resonances. Here, David speculated that the fluctuations would follow a chi-
squared distribution with one or two degrees of freedom. This is borne out
experimentally and is now considered one of the hallmark properties of the
compound nucleus. It also became a part of random matrix theory in
mathematical physics. Unlike the early parts of the thesis, David never
published the parts on compound-nucleus decay widths. However, physicists at
the Harwell Laboratory knew about David’s results and J.E. Lynn explained them
in his book [11]. Unfortunately, David’s treatment of fluctuations was not
recognized until very recently [12] and the distributions are known today
under other author’s names [13].

Figure 3: David and his children (left to right), Barbara, Thomas, and Anne-
Katherine 1969.

##  3 Research areas

David’s interactions with the physicists mentioned earlier were reflected not
only in David’s thesis but also in his early publications. One paper [14],
which dealt with angular momentum couplings and angular distributions of
$\gamma$-rays and other particles, is still the ”Bible” most experimentalist
use when they analyse their data, as we have been told by Peter Butler (FRS
2019) (Liverpool) and Yorick Blumenfeld (Orsay), and others. Early in his
research career David wrote the textbook Angular Momentum [15] with Ray
Satchler. This textbook was prominent among several texts published in this
time period. It was widely used by graduate students and post-graduates
working in nuclear theory. David also published a book on Nuclear Forces [16].

###  3.1 Effective interactions and calculations tools

In his thesis David had laid the basis for the use of effective interactions
in the calculations of matrix elements for nuclear structure studies. The idea
was greatly advanced in three later papers. The first proposes a gaussian form
for the effective nucleon-nucleon interaction known as the ”Brink-Boeker”
interaction [17] that all nuclear physicists have used at least once in their
lives. This paper was very influential at the time and was later developed by
Gogny and collaborators in the interaction that is widely used even today [18,
19].

In 1959 Tony Skyrme proposed modelling the effective interaction between
nucleons in nuclei by a short-range potential, an idea which is useful in
nuclear structure and the equation of state of neutron stars [20]. The Skyrme
force is an effective interaction depending on a small number of parameters
whose strength could be fitted to reproduce various bulk properties of nuclei
as well as selected properties of some nuclei, especially the doubly magic
nuclei. At the beginning of the 1970s David was a frequent visitor to the
Theoretical Division at the Institut de Physique Nucléaire, Orsay where his
sixty-fifth birthday was celebrated (figure 4). The work done there produced
two papers with Dominique Vautherin [21, 22] which were the basis for the
intense use of the so-called Skyrme interactions, in all their many present
variants. The papers revived a general interest in using Skyrme’s
parametrization of the nucleon-nucleon interaction to calculate nuclear
binding energies, and later to other aspects of nuclear structure. In effect,
the interaction is treated as an energy-density functional theory in the
spirit of the Kohn-Sham theory in condensed matter physics.

The Hartree-Fock calculations in [21] for spherical nuclei used Skyrme’s
density dependent effective interaction. This seminal paper showed how the
Skyrme force could be used to make accurate calculations of certain nuclear
properties and Vautherin and Brink developed these ideas further in a series
of papers which had a strong impact on nuclear structure calculations. T.
Otsuka comments: “The paper [21] has had a huge impact, as verified by the
number of citations $>$2000\. In nuclear theory, papers having the citation
index $>$1000 are rather few, which implies how important the Vautherin-Brink
paper is. This year is the 50 year anniversary of this paper, and it is
amazing that the basic formulation within the mean-field approach has not
changed too much, implying that the scheme presented in this paper is so
solid”.

The calculations of Vautherin and Brink were extended by many other physicists
during the subsequent period. In particular at Oxford, Micky Engel, Klaus
Goeke and Steve Krieger, together with Dominique Vautherin derived the energy
density using a Slater determinant where the single particle states were no
longer invariant under time reversal, as it is in the Hartree-Fock method.
With the Skyrme interaction the TDHF approach leads to an equation of
continuity for the single particle density [22]. This paper showed how Dirac’s
time-dependent Hartree-Fock theory could be applied to nuclear dynamics in a
light nucleus. In the year immediately following the publication, the theory
was applied to collisions involving a large number of nucleons [23], showing
that the method would be a powerful one for heavy nuclei as well. The method
is justified as a time-dependent density-functional theory, and it remains in
widespread use.

In 1973 Ica Stancu came to Oxford as a post doctoral fellow and worked with
David on heavy ion reactions in deriving the interaction potential of two 16O
nuclei starting from the Skyrme energy density formalism [24]. They included
the previously ignored tensor part of the Skyrme interaction. Along with an
additional effort from Hubert Flocard at Orsay, the Skyrme HF calculations
yielded single particle levels of spherical closed nuclei [25]. The role of
the tensor force is to contribute to the spin-orbit splitting of the single-
particle levels. For spherical closed shell nuclei the effect turned out to be
small. Later it was found that in spherical spin unsaturated nuclei it makes a
dramatic difference, giving the correct order of single particle levels, as,
for example, in the Sn isotopes [26]. Many experiments on neutron-rich nuclei
since 2006 have shown that the Skyrme formalism including the tensor force was
the simplest way to describe the shell evolution of neutron-rich or proton-
rich nuclei and indicated new magic numbers.

###  3.2 Heavy-ions and Semi-classical methods in Nuclear Physics

As tandem accelerators and cyclotrons were built to study heavy-ion Physics,
David started an intense collaboration with the experimentalists at the
Department of Nuclear Physics in Oxford. The accelerators were used to study
heavy-ion elastic scattering and direct reactions such as transfer and measure
masses and perform spectroscopy of neutron-rich matter. In those years
semiclassical methods were widely used in the Nuclear Physics community to
analyse data. They were particularly appropriate for heavy ions because of the
high incident energies and the large impact parameters involved. Thus David
started the Oxford school on the subject, more or less parallel in time to the
Copenhagen school of Broglia and Winter and collaborators. At that time, these
heavy-ion reactions were analyzed through the partial wave expansions of the
colliding partners, a methodology that was computationally demanding and
giving little insight to the underlying dynamics. David’s semi-classical
treatment of the collision was much simpler. Some of the early papers on the
theory of peripheral reactions were based on his student’s thesis, including
Hashima Hasan and Luigi Lo Monaco [27, 28].

David’s investigation of the kinematical effects in such reactions, for which
there was concrete experimental evidence from the work of Peter Twin (FRS
1993) and his collaborators at Liverpool, became a key element for
experimentalists. In the paper by the title ”Kinematical effects in heavy-ion
reactions” [29] David introduced a ”semi-classical amplitude” [30] that could
be used in DWBA-like calculations of transfer [31] and proposed a matching
condition to predict a large reaction cross-sections, a condition that was
beautifully adapted to understand spin-polarization experiments. He showed
that energy and angular momentum couplings in heavy-ion reactions led to very
selective matching rules by which high angular momentum single-particle states
could be populated. High angular momentum single-particle states sometimes
appear as low-lying continuum resonances. They have been studied by the method
of transfer-to-the-continuum [32] which has helped disentangle single-particle
from collective degrees of freedom and has also been applied in the so called
”surrogate reactions” as a substitute for free neutron beams.

Semi-classical ideas have been helpful in studying breakup and dissociation of
weakly bound radioactive ions including halo nuclei and other such unstable
nuclei whose dynamics is rather involved and difficult to study experimentally
due to the very low intensity of beams. David, Angela Bonaccorso and her
students got heavily involved in this new physics from the ’90s on, with a
long series of papers (see [33] and references therein), conference
contributions, meeting organization, some of them at the ECT* in Trento,
spanning the last forty years of David’s career. Finally it has recently been
shown [34] that the semi-classical treatment of breakup by David and his
collaborators is fully consistent with a quantum mechanical treatment.

David studied microscopic models for the real and imaginary parts of the ion-
ion optical potential to be used in elastic scattering calculations with Ica
Stancu. He also studied fusion with Neil Rowley and N. Takigawa. David and
Takigawa developed a semi-classical reaction theory with three classical
turning points which explained the anomalous large angle scattering of
${\alpha}$ particles as a quantum-mechanical interference between the barrier
wave and the internal wave, thereby providing an intuitively clear picture of
a complex phenomenon underlying nuclear reactions in terms of classical and
quantum ideas. David, Vautherin, and M.C. Nemes studied the effect of
intrinsic degrees of freedom on the quantum tunnelling of a collective
variable. This work was further developed by other theorists including Kouichi
Hagino who studied the deviation from adiabaticity in quantum tunnelling with
many degrees of freedom.

David met Uzi Smilanski in Munich when they were both there on sabbatical.
Both had worked on semi-classical approximations and gave a joint series of
lectures on this topic. David was concerned that the standard WKB method was
insufficient to explain tunnelling through a barrier and was particularly bad
near the barrier top. David and Uzi applied the uniform semi-classical method
evolved by Michael Berry (FRS 1982) to successfully address the problem [35].
Uzi remembers David as a physicist with excellent intuition and an ability to
grasp the essence of a problem before cracking the problem with rigorous
mathematics and complex computation.

David, Massimo di Toro, and Alberto Dellafiore developed a semi-classical
description of collective responses with a mean field approach paving the way
for a study of the dynamics of a nuclear Hartree-Fock fluid. When the national
heavy-ion laboratory started in Catania (LNS-INFN) around an advanced
superconducting cyclotron, David was a reference point for simple physics
suggestions.

###  3.3 Path integral methods in Nuclear Physics

David’s expertise with semi-classical methods for tackling quantum problems
naturally led him towards the Feynman path integral approach to quantum
mechanics which was based on a Lagrangian approach. Hans Weidenmüller had met
David at various conferences in the 1950s and 1960s and spent 1977-78 on a
sabbatical in Oxford. During this period David and Hans worked on the
application of the Feynman path integral method to the study of the heavy-ion
reactions and developed the Influence Functional approach to this problem
which David and his collaborators later used to establish master equations.
Hans remembers that at a summer school a few years later David delivered a
series of lectures on nuclear reactions. In the first lecture he developed the
topic using a dozen transparencies and in subsequent lectures used the same
transparencies in a different order to display and illuminate aspects of the
topic that had gone unnoticed before. Hans remembers it as a display of the
combination of simplicity and depth that were hallmarks of David’s approach to
Physics.

The path integral method was particularly well suited for studying problems
with many degrees of freedom in which classical description in terms of
trajectories was good for some degrees of freedom but not for all. Coulomb
excitation in heavy-ion collisions is an example where the relative motion of
the ions could be described in terms of coulomb trajectories but the
excitation of the quantum states of the ions had to be treated using quantum
mechanics. David and Sukumar [36] used the Feynman path integral method to
evolve a systematic way of arranging the correction terms for the quantum
amplitudes for processes involving coupled degrees of freedom where the
description in terms of classical trajectories was good for some degrees of
freedom. David, Sukumar, and Fernando Dos Aidos used this method to provide
corrections to the primitive semi-classical amplitude for Coulomb excitation
of heavy-ions. Sukumar and David used the path integral method to describe
spin-orbit coupling effects and together with Ron Johnson at Surrey and his
group successfully explained the experimental data on polarization effects.

Figure 4: David and his wife Verena, May 2018.

##  4 Other topics

David was very quick at grasping the core of a Physics problem and putting it
in simple, calculable terms. Often the problem required somewhat involved
analytical calculations, but he was a master of that. Thus anytime a visitor
went to Oxford with a new problem, David would start a very successful line of
research which he often followed up with his graduate students.

###  4.1 Cluster models

It happened for example with the cluster model physics, starting with the
seminal paper [37]. This paper developed the generator coordinate method of
Hill and Wheeler [38] to produce a practical tool to reduce the many-particle
Hamiltonian to an ordinary Schrödinger equation for a collective variable.
Thus the nuclear cluster model was related to the shell model. To treat
nuclear states in such different circumstances, a formulation which includes
clustering at one extreme and shell structure at the other extreme was needed.
David proposed microscopic multi-${\alpha}$-clusters treating four nucleons
with different spin-isospin states as a single particle orbit. Under anti-
symmetrisation of nucleons the cluster model wave-functions approximate shell
model functions and enabled the description of both cluster and shell model
structures in a unified way. Their approach was adopted and is in widespread
use even in present-day nuclear theory. The main applications up to now are on
spectroscopy and large-amplitude collective motion.

Y. Suzuki’s work on the cluster model was largely inspired by David’s paper on
”Do alpha clusters exist in nuclei?” [39] presented at a meeting in Tokyo in
1975. This paper contained all the essential components needed in the alpha
particle model, the microscopic theory beyond the shell model description
based on many-particle many-hole excitations, the relation between the
resonating group method GCM, the equilibrium arrangement of clusters,
extension of the Hill-Wheeler method, the angular momentum projection, and the
Slater determinant technique for evaluating matrix elements. Suzuki remembers
that David never forgot to mention that the original model was proposed by H.
Margenau and C.Bloch [40, 41, 42].

At the Varenna School in 1955 David met S. Yoshida from Japan and they
discussed inelastic scattering of protons and neutrons by deformed nuclei. By
chance David had a chapter in his thesis on this topic and Yoshida had been
studying the same subject. This interaction with Yoshida helped David to
develop strong connections with nuclear theory groups in Japan over many
years.

###  4.2 Bose-Einstein condensation of atoms

During his period as Deputy Director of ECT* in Trento, 1993-1998, David
interacted with many members of the Physics Department in Trento. One such
interaction with Sandro Stringari led to David’s interest in Bose-Einstein
condensation of alkali atoms in magnetic traps [43]. Sukumar and David [44]
developed an approximate method for calculating the rate of escape from the
magnetic trap thereby enabling an estimation of the duration for which the
condensate atoms can be held in the trap as a function of the ultra-cold
temperature and the strength of the magnetic field.

###  4.3 Miscellaneous

David was interested in the role of pairing interaction in finite nuclei and
this led to the study of nuclear superfluids. His book with R. Broglia [45] is
considered to be a wonderful exposition of this subject. David’s knack for
explaining detailed Physics in a simple and clear manner is abundantly evident
in this book. In the 1990s Ica Stancu raised David’s interest in the quark
structure of exotic hadrons named tetraquarks, a system of two quarks and two
antiquarks, and studied the stability of such systems containing heavy
quarks/antiquarks in a QCD inspired quark model. Even though David had not
worked on the Interacting Boson Model (IBM) he nevertheless provided
supervision for doctoral students such as Martin Zirnbauer who chose topics in
this field. He also supervised Hans Peter Pavel’s thesis on Schwinger pair
production in a flux tube model containing a chromomagnetic field.

##  5 Teaching and administrative roles

David’s doctoral students remember him for the gentle way he corrected them
when they had made errors. Many of the students learned from him how to take a
critical approach to their results and how it is possible to look at a complex
problem from several different viewpoints and find the one that gives the best
physical insight. They also remember the immense support he gave to their
research and pastoral care. Many graduate students also remember how much they
had learned from the courses he taught at Oxford and at Summer schools. His
book with Satchler [15] and paper with Rose [14] on angular momentum algebra
were found to be of immense value in formulating and tackling problems in
Nuclear Physics. Many researchers and students who met David were astonished
that someone with such towering achievements could be so humble, nice and
honest. David was very open-minded and we report a number of episodes to
illustrate this aspect of his character.

Future Nobel laureate Prof. Tony Leggett remembers: ” My undergraduate major
at Balliol was in Greats (classical languages, ancient history and philosophy)
and I was set to graduate (and eventually did so) in the summer of 1959.
Towards the end of the academic year 1957-1958, partly encouraged by the post-
Sputnik cultural swing towards science in the UK, I conceived the ambition of
taking a second undergraduate degree in physics and perhaps eventually making
my career in academia in that subject. Given that I had essentially no
meaningful exposure to physics at the high-school level and only a brief and
informal exposure to any kind of mathematics beyond simple differential
calculus (I’m not sure that I had even had that), such a drastic change of
academic direction was extremely unusual, indeed at the time almost unheard-
of. My first concern was to find a higher education institution which would
accept me for it and I rapidly concluded that my only hope was to apply to my
existing Oxford college, Balliol. David had just recently become the college’s
first tutor in theoretical physics (most Oxford colleges did not have such a
thing in 1958), so it fell on him to take the decision on my application. To
this end he asked me to read over the summer vacation a few chapters from the
book ”What is Mathematics?” by Courant and Robbins [46], perhaps the most
beautiful presentation I have ever seen of mathematical topics for the
layperson. When I returned to Oxford in the Fall of 1958 he gave me an
informal mini-exam on that material, and on the basis of my performance
decided to recommend to Balliol to accept me. In the event I did my physics
degree at Merton, who offered me a scholarship, but since they did not at the
time have a tutor in theoretical physics David played that role for me for
much of the two years which it took me to complete the degree. I think it is
virtually certain that had he made the opposite decision, I would never have
had a career in physics, and I am profoundly grateful to him for the
imagination he showed in going beyond my formal academic qualifications.”

Another story comes from Paul Stevenson: ”I was called up for interview at
Balliol in December 1991. The office I was in for that interview was David
Brink’s office, above the Senior Common Room. In the interview were me, David
Brink, David Wark, Jonathan Hodby (those three there for physics) and Bill
Newton-Smith (for philosophy). I don’t remember all the questions. I do
remember that David Brink showed me a postcard and asked me what, physically,
was wrong with the picture. It was a Japanese style print with a mountain in
the background and a lake in the foreground. There was a reflection of the
mountain in the lake, but it was off to one side. I saw what was wrong, and
struggled to articulate it in the language of a physicist, and in the end
David prompted me by asking what is particular about an incident light ray, a
reflected light ray, and the normal to the surface at which it is reflected
and I said the right thing - that they are all in the same plane. I was duly
accepted to Balliol and spent three years there studying physics”. Danny
Chapman remembers: ”I don’t think I’ll ever forget the ”sense” of David
Brink’s tutorials, and of being in the presence of such a sharp and insightful
mind. I remember being quite inspired once when my fellow student had tried to
answer a question in what I thought was an odd and probably wrong way, ending
up with a sum, which he then attempted to turn into an integral, which didn’t
work out. Rather than saying ”don’t do it like that, do it like this”, David
was able to continue from there and make it work, which was a really positive
experience and encouragement to follow every path to its end. I feel lucky to
have been at Balliol when he was there.”

Angela Bonaccorso remembers daily life as one of David’s students: At the
Department of Theoretical Physics there was a coffee room where coffee was
served between 11:00 and 11:30. We would try to be there on time to sit around
David who would be chatting with other senior members of the department or
some visitor. There would always be someone bringing up some interesting and
challenging new problem. Everyone gave an opinion, the atmosphere was
competitive. Most of the time David would win the argument and his students
felt very proud.Not all supervisors were so nice, helpful, and respectful of
us as David was. But it was not at all easy to be David’s student. First of
all we needed to have detective skills. David was very busy and very elusive.
In those days there was no email or SMS. The only way to be sure that he was
inside was to look for his bicycle. If the bicycle was outside we would knock
at the door of his office and if we were lucky he would answer and let us in.
In spite of all his many commitments we always managed to have at least one
chat per week with him. Another reason why it was not easy to be his student
was that David had a very original way of understanding things and finding the
way out of problems. During our conversations often he would stop talking and
be silent for five to ten minutes, rubbing his hand on his forehead. Then he
came up with some equation, or a drawing or something like that and he would
tell us: I think it is like this…I think we should get something like
that…etc. I (we) would stare at him speechless and in wonder. Where did the
’oracle’ come from? Most of the time this was the end of the meeting. I (we)
left his office rather puzzled, worked desperately hard for one week and if we
had managed to understand his line of thought, after pages and pages of
calculations, we would find exactly what he had predicted. We all knew it was
like that, we all passed this information on to each other, generation after
generation: listen to David, he is always right, just try to reproduce the
miracle of his craftsmanship in physics.

A further proof of how much busy David was and how precious was for everyone
the time spent in conversation with him can be found in the comment Gerry
Brown made in his review for Science [47] of the Proceedings of the Varenna
summer school [41] : ’Let me draw special attention also to the article of
David Brink, ”The alpha-particle model of light nuclei,” which is one of the
most beautiful developments in this subject. Brink likes to sit on his work
for years and, on the whole, doesn’t even answer letters inquiring about it,
so that one must either adopt the expedient of traveling to Oxford to talk
with him, or invite him to lecture at summer schools. Both are worth while.’

David was a pillar of Balliol college and Department of Theoretical Physics
for decades, an immensely popular tutor and supervisor, a cheerful and always
helpful colleague, and a wonderful guide to younger colleagues and
administrative staff who happened to be working with him. David had another
long and distinguished career in Italy after he left Oxford. Following an
invitation from Renzo Leonardi he moved to Trento as full professor of History
of Physics and helped in establishing the ECT*, European Center for
Theoretical Studies in Nuclear Physics and Related Areas. The Nobel laureate
Ben Mottelson was the founding director and David the vice-director, while
Renzo Leonardi was the Scientific Secretary. In the five years David spent at
Trento he took care of organising various technical aspects of the secretarial
offices, library, computer center and visitor hospitality. At the same time he
gave very productive contributions to workshops with his constant presence,
his huge knowledge of nuclear physics and stimulating discussions. The superb
reputation and international standing of this extremely important European
initiative is undoubtedly due in large part to David’s wisdom in its crucial,
formative years.

Figure 5: David’s sixty-fifth birthday celebration. Orsay, 1995.

##  6 Career, Honours and Awards

1954-55 Royal Society Rutherford Scholarship.
1957-1958 Instructor at the Massachusetts Institute of Technology (MIT).
1958 Fellow of Balliol College and Lecturer in Theoretical Physics, Oxford.
1976-1978 Vice-Master of Balliol College.
1981 Fellow of the Royal Society.
1982 Rutherford Medal of the Institute of Physics.
1988 H. J. G. Mosley Reader at Oxford.
1990-1993 Senior Tutor, Balliol College, academic planning and administration,
Oxford.
1992 Foreign member of the Royal Society of Sciences, Uppsala.
1993-1998 ECT*, Trento, Vice-Director .
1993-1998 Full professor of History of Physics, University of Trento.
2006 Varenna Conference on Nuclear Reactions dedicated to him.
2006 Lise Meitner prize of the European Physical Society shared with H. J.
Kluge.

Visiting scientist at :

  * •

Niels Bohr Institute 1964,

  * •

University of British Columbia 1975,

  * •

Institut de Physique Nucléaire d’Orsay 1969 and 1981-1982,

  * •

The Technical University of Munich 1982,

  * •

University of Trento 1988,

  * •

University of Catania 1988,

  * •

Michigan State University 1988-1989.

##  7 Acknowledgements

The authors are greatly indebted to the Brink family for sharing with them
private memories and photographs and for a critical reading of the manuscript.
A large number of friends and colleagues, too many to be individually
mentioned, contributed with their appreciation of David’s life and scientific
career. Ica Stancu and Sharon McGrayne Bertsch read and commented the
manuscript. One of us (AB) gratefully acknowledges George F. Bertsch for his
help in digging out from David’s thesis and early work the roots of several
founding pillars of modern Nuclear Physics.

## References

  * [1] Elliott J.P., Lane A.M. 1957 The Nuclear Shell-Model. Structure of Atomic Nuclei / Bau der Atomkerne. Encyclopedia of Physics / Handbuch der Physik vol 8 / 39. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45872-9_4
  * [2] Brink, D. M., 1955 https://ora.ox.ac.uk/objects/uuid:334ec4a3-8a89-42aa-93f4-2e54d070ee09.
  * [3] Hughes, D. J., and Harvey, J. A. 1954 Radiation-widths of nuclear energy-levels Nature 173, 942 - 943. DOI: https://doi.org/10.1038/173942a0
  * [4] Feshbach, H. Porter, C.E. and Weisskopf, V.F. 1954 Model for Nuclear Reactions with Neutrons Phys. Rev. 96 448-464. DOI:https://doi.org/10.1103/PhysRev.96.448
  * [5] Capote, R. et al., 2009 RIPL, Reference Input Parameter Library for Calculation of Nuclear Reactions and Nuclear Data Evaluations Nuclear Data Sheets 110 3107-3214. DOI: https://doi.org/10.1016/j.nds.2009.10.004
  * [6] Axel, P. 1962 Electric Dipole ground-state transition width strength function and 7 Mev photon interaction Phys. Rev. 126, 671-683. DOI:https://doi.org/10.1103/PhysRev.126.671
  * [7] Brink, D. M., 1957 Individual Particle and Collective Aspects of the Nuclear Photoeffect Nucl. Phys. 4  215 - 220. DOI: 10.1016/0029-5582(87)90021-6
  * [8] Goldhaber, M. and Teller, E. 1948 On Nuclear dipole vibrations Phys. Rev. 74 1046 - 49. DOI:https://doi.org/10.1103/PhysRev.74.1046
  * [9] Steinwedel, H., Jensen, J. H. D. and Jensen, P. 1950 Nuclear dipole vibrations Phys. Rev. 79 1019\. DOI:https://doi.org/10.1103/PhysRev.79.1019
  * [10] Newton, J. O. et al. 1981 Observation of Giant dipole resonances built on states of high energy and spin Phys. Rev. Lett. 46 , 1383-1386. DOI:https://doi.org/10.1103/PhysRevLett.46.1383
  * [11] Lynn, J.E., 1968 Theory of neutron resonance reactions, (OUP) 321.
  * [12] Hagino, K. and Bertsch, G. F., 2021, Porter-Thomas fluctuations in complex quantum systems Phys. Rev. E104 L052104. DOI:https://doi.org/10.1103/PhysRevE.104.L052104 and references therein.
  * [13] Porter, C.E. and Thomas, R.G. 1956 Fluctuations of Nuclear Reaction widths Phys. Rev. 104, 483-491. DOI:https://doi.org/10.1103/PhysRev.104.483
  * [14] Rose, H.J. and Brink, D.M. 1967 Angular Distributions of Gamma Rays in Terms of Phase-Defined Reduced Matrix Elements Rev. Mod. Phys. 39 , 306-347. DOI: 10.1103/RevModPhys.39.306
  * [15] Brink, D. M. and Satchler, G. R., 1962 Angular Momentum, (OUP).
  * [16] Brink, D. M. 1965 Nuclear Forces, (Pergamon).
  * [17] Brink, D.M. and Boeker, E. 1967 Effective interactions for Hartree-Fock calculations Nucl. Phys. A 91, 1-26. DOI: 10.1016/0375-9474(67)90446-0
  * [18] Gogny, D., Pires, D. P. and De Tourreil, R. 1970 A smooth realistic nucleon-nucleon force suitable for nuclear Hartree-Fock calculations Phys. Lett. B32 591-595. DOI: https://doi.org/10.1016/0370-2693(70)90552-6
  * [19] Decharge, J. and Gogny, D.1980 Hartree-Fock-Bogolyubov calculations with the D1effective interaction on spherical nuclei Phys. Rev. C21 1568-1593. DOI:https://doi.org/10.1103/PhysRevC.21.1568
  * [20] Skyrme, A. 1959 The effective nuclear potential Nucl. Phys. 9 615-634. DOI:https://doi.org/10.1016/0029-5582(58)90345-6
  * [21] Vautherin , D. and Brink, D. M.1972 Hartree-Fock calculations with Skyrme’s interaction. 1. Spherical nuclei Phys. Rev. C5 626-647. DOI: 10.1103/PhysRevC.5.626
  * [22] Engel, Y.M., Brink, D. M., Goeke, K., Kriege, S.J. and Vautherin, D. 1975 Time dependent Hartree-Fock theory with Skyrme’s interaction Nucl. Phys. A249, 215-238. DOI: 10.1016/0375-9474(75)90184-0
  * [23] Bonche, P., Koonin S., and Negele, J. W., 1976 One-dimensional nuclear dynamics in the time-dependent Hartree-Fock approximation Phys. Rev. C13 1226-1258. DOI:https://doi.org/10.1103/PhysRevC.13.1226
  * [24] Brink, D.M.and Stancu, Fl.1975 Interaction potential between two O-16 nuclei derived from the Skyrme interaction Nucl. Phys. A 243 175-188.
  * [25] Stancu, Fl., Brink, D.M. and Flocard, H. 1977 The tensor part of Skyrme’s interaction Phys. Lett. B68 108-112.
  * [26] Brink, D.M. and Stancu, Fl., 2007 Evolution of nuclear shells with the Skyrme density dependent interaction Phys. Rev. C 75 064311\.
  * [27] Hasan, H. and Brink, D. M. 1979 The transfer amplitude and angular distributions in heavy-ion reactions J. Phys. G: Nucl. Phys. 5 771\.
  * [28] Lo Monaco, L. and Brink, D. M. 1985 Perturbation approach to nucleon transfer in heavy-ion reactions J. Phys. G: Nucl. Phys. 11 935-952.
  * [29] Brink, D.M. 1972 Kinematical effects in heavy-ion reactions Phys .Lett. B40 37-40. DOI: 10.1016/0370-2693(72)90274-2
  * [30] Brink, D. M., 1985. Semi-Classical Methods in Nucleus-Nucleus Scattering, (Cambridge University Press).
  * [31] Anyas-Weiss, N., Cornell, J.C., Fisher, P.S. ,Hudson, P.N., Menchaca-Rocha, A., Millener, D.J., Panagiotou, A.D., Scott, D.K., Strottman, D., Brink, D.M., Buck, B., Ellis, T.P. and Engeland, J., 1974 Nuclear structure of light nuclei using the selectivity of high energy transfer reactions with heavy ions Physics Reports 12 201-272. DOI: 10.1016/0370-1573(74)90045-3
  * [32] Bonaccorso, A. and Brink, D. M., 1988 Nuclear transfer to continuum state Phys. Rev. C38 1776-1786; 1991 Stripping to the continuum of 208Pb Phys. Rev. C44 1559-1568.
  * [33] Bonaccorso, A. and Brink, D. M., 2021 Models of breakup: a final state interaction problem Eur. Phys. J.A57 171\.
  * [34] Jin Lei, Bonaccorso, A. 2021 Comparison of semiclassical transfer to continuum model with Ichimura-Austern-Vincent model in medium energy knockout reactions Phys. Lett. B813 136032\. DOI:https://doi.org/10.1016/j.physletb.2020.136032
  * [35] Berry, M. V. 1966 Uniform approximation for potential scattering. Proc. Phys. Soc. 89, 479-490.
  * [36] Sukumar, C. V. and Brink, D.M. 1983 Path integral methods for inelastic scattering Nucl. Phys. A404 121-141.
  * [37] Brink, D.M. and Weiguny, A. 1968 The generator coordinate theory of collective motion Nucl. Phys. A120 59-93. DOI: 10.1016/0375-9474(68)90059-6
  * [38] Hill, D. L. and Wheeler, J. A. 1953 Phys. Rev. 89, 1102.
  * [39] Brink, D. M. 1975 Do alpha-clusters exist in nuclei. Proceedings of the INS-IPCR Symposium on Cluster Structure of Nuclei and Transfer Reactions Induced by Heavy-Ions, Tokyo, March 17-22, 1975.
  * [40] Margenau, H. 1940 Phys. Rev. 5, 37.
  * [41] Brink, D. M. 1966 The Alpha-Particle Model of Light Nuclei, Scuola Internazionale di Fisica ’Enrico FERMI’, XXXVI Corso, Ed. C. Bloch, 247.
  * [42] Brink, D. M. 2008 J. Phys.: Conf. Ser. 111, 012001.
  * [43] Brink, D.M., Stringari, S., 1990 Density of states and evaporation rate of helium clusters. Z Phys D-Atoms, Molecules and Clusters 15 257-263 .
  * [44] Sukumar, C. V. and Brink, D.M. 1997 Spin flip transitions in a magnetic trap Phys. Rev. A56 2451-2454.
  * [45] Brink, D. M. and Broglia, R. A., 2005 Nuclear Superfluidity Pairing in Finite Systems, (CUP).
  * [46] Courant, R. and Robbins, H. 1941 What is mathematics?  Oxford, UK: Oxford University Press.
  * [47] Brown, G. E., 1967 Nuclear Physics: Many-Body Description of Nuclear Structure and Reactions. Course 36, International School of Physics ’Enrico Fermi.’ C. Bloch, Ed. Academic Press, New York, 1966. Science, 158 (3807), DOI: 10.1126/science.158.3807.1440