# Projection: A Mixed-Initiative Research Process

Austin Silveria
California Polytechnic State University, San Luis Obispo
San Luis Obispo, CA 93405
<EMAIL_ADDRESS>

###### Abstract

Communication of dense information between humans and machines is relatively
low bandwidth. Many modern search and recommender systems operate as machine
learning black boxes, giving little insight as to how they represent
information or why they take certain actions. We present Projection, a mixed-
initiative interface that aims to increase the bandwidth of communication
between humans and machines throughout the research process. The interface
supports adding context to searches and visualizing information in multiple
dimensions with techniques such as hierarchical clustering and spatial
projections. Potential customers have shown interest in the application
integrating their research outlining and search processes, enabling them to
structure their searches in hierarchies, and helping them visualize related
spaces of knowledge.

##  1 Introduction

We consider conceptualizing the online research process as a specific instance
of the scientific method: posing a question, collecting information to answer
the question, organizing the information, and communicating the answer, where
the steps are cycled multiple times in the process.

In interviews conducted by the author, we talked to five college students
about their research process. We learned that they begin the research process
with an outline of how they think about a topic. They then search Google with
targeted queries to find information to support this structure. Along the way,
their outline evolves and prompts new searches. We learned that people are
frustrated by the lack of integration between their outline and searches.
_I’ve organized my ideas in this outline, but how do I communicate that to
Google? How can I search for new information in the context of research I’ve
already done?_ We learned that people find it difficult to survey a topic at a
high level. _I have this idea of the field that I’m trying to explore, but how
can I map that field into subtopics to understand what makes it up? I find a
number of articles through my searches, but it’s overwhelming to manually read
and organize all of them._

As an extension to our previous work [1], we built Projection, a mixed-
initiative interface to the research process. The tool clusters documents on a
map based on their similarity (as seen in Figure 1), the goal being that
similar documents show up on the map next to each other and can be grouped
into hierarchical clusters [2]. Projection also includes a function to search
on the map (as seen in Figure 2), project results in nearby space, and group
them into hierarchical clusters [3]. We demoed the functionality to the same
five interviewees to spur discussion about product development. We asked them:
_given this organized knowledge space, how would you want to navigate it? How
would you want to organize it yourself? What kind of information would you map
out? How is this type of search different from current solutions?_ We learned
that they would want to start typing on a map, visualize related topics, and
continue searching through those topics, building a hierarchy of what they are
thinking and continuing to search in the context of that hierarchy. They
mentioned a side tab with a traditional outline. One could click on the
outline to go to that place on the map. One could use the outline to maintain
a context of where they are in the knowledge space. We talked about multiple
people interacting with the same information in Projection. _How do other
people structure these topics I’m interested in? In the context of a large
organization, who should I talk to about this topic? Who has spent a lot of
time structuring this realm of information?_ We learned that they would want
to explore types of information such as wikipedia, research articles, web
articles, and personal documents. We found that Projection can be compared to
traditional search in the following way: instead of a list of pages,
Projection visualizes a knowledge space. Instead of searching with a single
natural language query, Projection searches with a hierarchy of natural
language. Instead of each search being independent, Projection searches come
together into one navigable structure.

Figure 1: Projection Spatial Document Clustering [2]

##  2 Backgound

###  2.1 Data Model

Consider an entity graph, where the meaning of an entity depends on its
relationship to and the meaning of entities connected to it. We apply this
data model to Projection such that each piece of information is an entity that
can be structured and searched for in the context of other entities. As an
example, consider a hierarchy of entities mapping the sub topics of a field.
The parent-child connections of the hierarchy communicate the relationship
between entities and can be used to find more entities that fit into the
structure.

###  2.2 Mixed-Initiative Interface

Mixed-initiative interaction refers to a flexible interaction strategy in
which each agent (human or computer) contributes what it is best suited at the
most appropriate time [4]. In applying this to Projection, humans manually
structure entities and the computer responds with contextually relevant
entities projected into the structure. For example, after creating an entity
with the language "investing", the machine might project clusters of entities
about types of investments, famous investors, or investment strategies. As
more structure is given to the entities by the human, the machine has more
context to suggest relevant types of entities.

###  2.3 Software Libraries

React [5] is a JavaScript library Projection uses for building declarative and
component-based interfaces. React automatically re renders components when
their inputs change, saving developers from manually triggering updates.

Redux [6] is a predictable state container for JavaScript applications. State
is interacted with via a global store exposed by Redux. React components
select subsets of this state to render and emit actions to update state based
on user interaction. When a component’s selected state changes, it is
automatically re rendered. With this functionality, Redux serves as a powerful
communication model between many declarative functions that need access to the
same state.

Mapbox [7] is a JavaScript library for building map applications in the
browser. It exposes a powerful map API that developers can interface with to
render components in 2D/3D space–the library then manages the entire map once
it is rendered: zoom, drag, re renders, etc.

Projection uses AWS Amplify [8] to manage web authentication and database
hosting. With a single GraphQL schema, Amplify can deploy a GraphQL server and
multiple DynamoDB tables to fully support queries and mutations in a managed
cloud environment.

On the backend we use Tensorflow’s Universal Sentence Encoder [9] to compute
language embeddings and Facebook’s FAISS [10] for large scale vector
operations such as similarity search.

##  3 Related Work

###  3.1 Data Model

The entity graph data model can be seen in various environments of
computationally defining meaning: the meaning of a word is the meaning of
words around it [11], the attention computation is a fully connected, directed
graph with learned connection weights [12], the meaning of an Instagram
account is the accounts it interacts with [13], and the meaning of a YouTube
video is the context in which it was watched [14].

People also consciously organize meaning with discrete connections. A
hierarchical research outline is a tree of natural language, document
citations form a document graph, and digital friends lists connect us to
others online. Unconsciously, a perspective could be that our brains define
meaning by relating information in multiple dimensions. Visual perception
relates points of space based on position and color. Auditory perception
detects changes in pressure of the environment across time. Our brain relates
information across all of these dimensions unconsciously, and we use this high
dimensional perception to act.

Work done by Yang et al. [15] explores the value of decoupling the learning of
entity relationships and semantic meaning. Compared to self-attention, which
fuses the computation of entity structure and meaning into one network, their
work decouples the two, allowing the learned entity structure to transfer to
new tasks where different features may be needed to model semantic meaning.
This is interesting from our perspective since Projection aims to be a mixed-
initiative interface that allows people to structure information in any
format. Structuring information in different ways or structuring different
types of information may be analogous to different tasks, warranting
consideration of different semantic features. Structures created by everyone
using Projection can also be valuable training data for computing entity
meaning.

###  3.2 Mixed-Initiative Interface

Examples of mixed-initiative interfaces include recommender systems of popular
internet applications. People interact with social connections online and
Instagram recommends new social connections [13]. People watch YouTube videos,
and YouTube learns from this context to suggest new videos [14]. People search
Google, and the model learns from this history to auto complete relevant
queries.

Figure 2: Projection Search [3]

Many existing mixed initiative interfaces are imbalanced–the machine learning
model operates in a higher dimension than the human guiding it, leading to a
lower bandwidth of communication between the model and human. It is difficult
to understand the model’s interpretation of meaning and guide its suggestion
making.

To increase the bandwidth of communication between the model and human, a
mixed-initiative interface can be designed with multiple dimensions: searching
based on a hierarchy of language, clustering search results to communicate
topics, projecting entities on a 2D/3D space to communicate their
relationships, or at the highest dimension with a brain-machine interface.
Even a small increase in dimension of the human side of the interface can
exponentially increase the bandwidth of communication, improving the shared
meaning representation between model and human.

##  4 System

We used AWS Amplify [8] to generate a set of DynamoDB tables to track object
state and a GraphQL server to support object queries and mutations. We define
a single GraphQL schema with our application’s data model and leverage Amplify
to manage the rest.

Outside of Amplify, we split the Projection architecture into two components:
a web interface to search, explore, and structure information and a backend to
support stateful computation of data such as document similarities, clusters,
and spatial entity projections. Our full architecture can be seen in Figure 3.

###  4.1 Interface

The interface supports two core actions: humans creating/structuring entities
and the model suggesting new entities/clusters in the context of that
structure. People drive the structuring of information and the model responds
with information that meshes with that structure.

To track application state, we define the data model shown in Table 1. We use
React [5], Redux [6], and Mapbox [7] to build modular interface components
that react independently to application state changes and mirror the data
model. The _Menu_ supports creating and selecting maps. The _Map_ component
reacts to a map being selected in the _Menu_ component, queries the map’s
entities, and renders multiple _Entity_ components on the map. The _Entity_
and _Document_ components query and render model data. Additional _Cluster_
components render sets of _Entity_ s or _Document_ s as hierarchical clusters.

This modular component architecture on top of React and Redux allow us to
easily test new components. A new component can be inserted anywhere in the
interface model, Redux can be leveraged to query state of other components and
communicate state updates, and we can let React handle efficient re renders
based on state changes.

Table 1: Projection Data Model Menu |  Map |  Entity |  Document
---|---|---|---
selectedMapId |  mapId |  entityId |  documentId
|  name |  mapId |  title
|  entityIds |  parentEntityId |  url
|  |  childrenEntityIds |  text
|  |  coordinates |
|  |  text |

###  4.2 Backend

At the core of Projection’s backend is FAISS [10], a library used to
efficiently store dense vectors and compute vector similarities. A web server
sits on top of FAISS and exposes _add_ and _query_ operations.

We organize Projection’s backend computation into a functional pipeline for
each operation. _add_ supports loading new language entities into FAISS by
first computing a vector representation of the language with the Universal
Sentence Encoder [9], then loading it into the index. The _query_ operation
receives an entity tree as input and executes the following pipeline to
construct a response: embed the text of each entity, search for
documents/entities in context of the tree, cluster the resulting
documents/entities, and project the response entities in space near the input
entity tree.

Figure 3: Projection Architecture

One strategy we consider to search in hierarchical context is as follows: find
neighbors of the root entity, find neighbors of children within the parent
level’s neighborhood space, and continue in a depth first search fashion. This
would allow human operators to search by "partitioning" the meaning space with
their structured entity hierarchy. With the functional pipeline architecture,
new strategies can easily be injected into the pipeline for testing.

To cluster entities represented by dense vectors in 2D space we first reduce
the dimensionality of the vectors from 512 to 2 with T-SNE, then apply
agglomerative clustering to find hiearchical structure.

Our goal for projecting entities is to move a set of entities in 2D space and
scale their distances while maintaining directional relationships. Our initial
multiplication transform is acceptable for testing, but future work should use
a known mathematical operation for maximum precision.

###  4.3 MVP

To create an initial MVP based on the architecture described above, we
gathered web pages from our personal search history and loaded them into the
backend. Web pages can be explored with hierarchical clusters on the map [2]
and searched for by typing on the map [3].

##  5 Evaluation

To evaluate our product, we demoed the functionality to five college students
as discussed in the introduction. All interviewees were excited by the
potential of the product, but had a different purpose in mind than a
traditional search engine. They would use Projection when they didn’t have a
specific answer in mind. They would want start typing on the map and explore
the different perspectives of their natural language query. They would connect
their initial query to new information, and continue their search/exploration
of the knowledge space.

For Projection to be usable, the interviewees highlighted a few points of
important functionality. First, it needs to contain the data that they are
interested in exploring. For those we interviewed, this was Wikipedia entries,
research articles, web articles, and personal documents. Filters should be
available to narrow the information considered to a certain type. They need to
be able to maintain context of where they are in the knowledge space. It was
easy to get lost on the map because parent clusters split into child clusters
on zoom, and weren’t visible for reference after. They mentioned being able to
see multiple levels of parent-child clusters on the map at the same time in
order to maintain depth context. They also mentioned a sidebar with a
traditional bullet point outline. This outline could represent their mapped
entity structure in a familiar format and highlight where they currently are
in the knowledge space. The outline would be interactive: clicks take one to
that location on the map, and the current location on the map is highlighted
in the outline. Another piece of requested functionality was group search. _I
have this group of entities here and I want to find more entities that fit in
the group_. In the future, this can easily be implemented with similarity
search and tested in our backend’s query pipeline.

##  6 Conclusion

We present Projection, a mixed-initiative interface to the research process.
Through hierarchical clustering, spatial projections, and an intuitive
knowledge structure, Projection can increase the bandwidth of communication
between humans and machines, a central issue in today’s interfaces to machine
learning. The human and machine represent information as an entity graph,
which can be used to communicate structure and meaning by both parties. Our
scalable software architecture will allow rapid experimentation of new
features and parallel development going forward.

##  7 Future Work

We have a strong roadmap of future work ahead based on our interview
discussions. Likely the most important step going forward is expanding the
types and scale of information accessible in Projection beyond our personal
search history. Implementing a web scraper to collect web pages can automate
this process, but customers should also be able to import their own documents
or trigger a scrape of a website. This would allow potential customers to test
the tool with information that is valuable to them, making their experience
and feedback more authentic.

Experimenting with the interface is another crucial step going forward. We
will need to discover the right amount of information to display at one time,
how to communicate where one is in their defined structure and the overall
knowledge space, and how to communicate the contents of varying types of
entity clusters. To conduct real tests, this should be done after additional
information is loaded into Projection so that potential customers can use the
product for practical purposes.

With the ability to conduct real tests, we will also be able experiment with
various machine learning methods on the backend. Testing different search
strategies and machine learning models will help us identify what solves
customer problems the best.

We will also continue abstracting our architecture with the goal of enabling
the lowest cost of experimentation. The cost of experimentation can be broken
into three parts: complexity of implementation, effort required to implement,
and time required to implement. A set of functions that react independently to
global state changes has proven to be a powerful architecture, one that we are
looking to adopt across our entire software stack. Functions in the frontend
or backend, across any number of physical machines, communicate via access-
controlled namespaces of a global state store and react to queried state
changes to produce additional state changes.

##  8 Acknowledgements

We would like to thank Foaad Khosmood for advising our research.

## References

  * [1] Austin Silveria.  Exploratory search with sentence embeddings.  CoRR, abs/2007.11198, 2020.
  * [2] Projection spatial clustering.  https://youtu.be/1-KDR6ohjCo.
  * [3] Projection search.  https://youtu.be/EOLx7NftanM.
  * [4] J.e. Allen, C.i. Guinn, and E. Horvtz.  Mixed-initiative interaction.  IEEE Intelligent Systems, 14(5):14–23, 1999.
  * [5] React: A javascript library for building user interfaces.  https://reactjs.org/.
  * [6] Redux: A predictable state container for js apps.  https://redux.js.org/.
  * [7] Mapbox: Maps and location for developers.  https://www.mapbox.com/.
  * [8] Aws amplify: Fastest, easiest way to build mobile and web apps that scale.  https://aws.amazon.com/amplify/.
  * [9] Universal sentence encoder.  https://tfhub.dev/google/universal-sentence-encoder/4.
  * [10] Faiss: A library for efficient similarity search and clustering of dense vectors.  https://github.com/facebookresearch/faiss.
  * [11] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean.  Efficient estimation of word representations in vector space, 2013.
  * [12] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  CoRR, abs/1706.03762, 2017.
  * [13] Ivan Medvedev, Haotian Wu, and Taylor Gordon.  Powered by ai: Instagram’s explore recommender system.  https://ai.facebook.com/blog/powered-by-ai-instagrams-explore-recommender-system/, 2019\.
  * [14] Paul Covington, Jay Adams, and Emre Sargin.  Deep neural networks for youtube recommendations.  In Proceedings of the 10th ACM Conference on Recommender Systems, New York, NY, USA, 2016.
  * [15] Zhilin Yang, Junbo Jake Zhao, Bhuwan Dhingra, Kaiming He, William W. Cohen, Ruslan Salakhutdinov, and Yann LeCun.  Glomo: Unsupervisedly learned relational graphs as transferable representations.  CoRR, abs/1806.05662, 2018.


# Properties of Higher-Order Prime Number Sequences

Michael P. May  443 W 2825 S
Perry, Utah
84302<EMAIL_ADDRESS>
###### Abstract.

In this paper, we analyze properties of prime number sequences produced by the
alternating sum of higher-order subsequences of the primes. We also introduce
a new sieve which will generate these prime number sequences via the
systematic selection and elimination of prime number indexes on the real
number line.

##  1\. Introduction

We provide a brief explanation of the concept of prime numbers with prime
subscripts. To this end, we refer the reader to the OEIS reference [1] on
higher-order prime numbers, and to the paper [2]. Higher-order prime numbers,
also called superprime numbers or superprimes, are the primes that occupy
prime-numbered positions within the sequence of all prime numbers. They are
also called prime-indexed primes. The OEIS sequence A006450 [4] defines this
sequence of prime numbers as

$a(n)=p_{p_{n}}=3,5,11,17,31,41,59,67,83,109,127,157,179,191,\dots\;.$

That is, if $p_{i}$ denotes the $i$th prime number, then the numbers of
sequence A006450 are of the form $p_{p_{i}}$.

Denote by $\mathbb{P}$ the sequence of primes $p_{1},p_{2},\dots$. Suppose
$A\subseteq\mathbb{P}$, and suppose $I$ is an index set of $A$, that is, for
every $n\in I$ we have $p_{n}\in A$. Then we define $A_{I}=\\{p_{n}\mid n\in
I\\}$. Note that the indices are the position in the whole sequence of primes
and not in $A$.

We define $P^{(k)}$ recursively by

$P^{(1)}=\mathbb{P}\text{ and }P^{(k)}=P_{P^{(k-1)}}\text{ for }k\geq 2.$

Thus $P^{(1)}$ are the primes, $P^{(2)}$ are the prime-indexed primes, and so
on. We use the notations $P^{(k)}$ and $p^{(k)}$ interchangeably.

DOI: 10.35834/YYYY/VVNNPPP

MSC2020: 11A41, 11B99

Key words and phrases: Primes, Prime number subsequences, Prime number

indexes, Infinite sequences, Alternating sum, Higher-order primes, Sieves

We will show that there is a unique partition of $\mathbb{P}$ into an index
set $I$ and an indexed set $\mathbb{P}_{I}$. We denote these by
$\mathbb{P}^{\prime}=I$ and $\mathbb{P}^{\prime\prime}=\mathbb{P}_{I}$. We
will see later that these two sets can be defined in terms of the sequences of
higher-order primes.

######  Theorem 1.1.

There is a unique subset $I$ of the prime numbers such that

$\mathbb{P}={I}\cup\mathbb{P}_{{{I}}}$

and

${I}\cap\mathbb{P}_{{I}}=\emptyset,$

###### Proof.

We begin by showing that ${I}$ is unique and that the first $n$ elements of
${I}$ and $\mathbb{P}_{{I}}$ determine the $n+1st$ element of ${I}$. First, we
write

$q_{1},\dots,q_{n}$

to designate the smallest elements of ${I}$ in increasing order. And because
the $nth$ prime function $p_{n}$ is a monotonic increasing function, it
follows that

$p_{q_{1}},\dots,p_{q_{n}}$

are the smallest elements of $\mathbb{P}_{{I}}$ in corresponding increasing
order. Now let $q$ be the smallest prime number that is different from all
$q_{1}$,…,$q_{n}$ and $p_{q_{1}}$,…,$p_{q_{n}}$. We know that either $q\in{I}$
or $q\in\mathbb{P}_{{I}}$, and our claim is that $q\in{I}$. But if we suppose
to the contrary that $q\in\mathbb{P}_{{I}}$, then there must be a prime number
$r$ such that $q=p_{r}$. Now we assumed that $q>p_{q_{n}}$, and it follows
from the definition that $r>q_{n}$ and $r<p_{r}=q$. But this means that $r$
satisfies the same conditions as $q$: i.e., that $r$ is the smallest prime
different from all $q_{1}$,…,$q_{n}$ and $p_{q_{1}}$,…,$p_{q_{n}}$. So we
should have chosen $r$ instead of $q$ because $r$ is smaller than the $q$ we
assumed to be an element of $\mathbb{P}_{{I}}$. Therefore, our assumption that
$q\in\mathbb{P}_{{I}}$ is a contradiction, and the smallest prime $q$
different from all $q_{1}$,…,$q_{n}$ and $p_{q_{1}}$,…,$p_{q_{n}}$ must be an
element of I because the elements of ${I}$, which serve as the subscripts of
$\mathbb{P}_{{I}}$, always lag behind the elements of $\mathbb{P}_{{I}}$.
Furthermore, since all prime numbers $\mathbb{P}$ are consumed in the process
of adding the smallest available prime number different from all
$q_{1}$,…,$q_{n}$ and $p_{q_{1}}$,…,$p_{q_{n}}$ to ${I}$, then we have shown
that ${I}$ and $\mathbb{P}_{I}$ are complement sets of $\mathbb{P}$. ∎

A new sieve will be introduced shortly which iteratively applies the smallest
available prime number as a subscript ${I}$ to the set of all prime numbers
$\mathbb{P}$ so that eventually all primes get used up and the partition is
created for $\mathbb{P}$.

##  2\. Alternating Sum

We discovered that the unique subset of prime numbers
$\mathbb{P{{}^{\prime}}}$ can be generated via an alternating sum of prime
number subsequences of increasing order. Thus, one can define
$\mathbb{P{{}^{\prime}}}$ as

$\mathbb{P^{{}^{\prime}}}={\left\\{{(-1)^{n-1}}\left\\{{p^{(n)}}\right\\}\right\\}}_{n=1}^{\infty}$
(2.1)

where the right-hand side of Eq. 2.1 is an expression of the alternating sum

$\left\\{{p^{(1)}}\right\\}-\left\\{{p^{(2)}}\right\\}+\left\\{{p^{(3)}}\right\\}-\left\\{{p^{(4)}}\right\\}+\left\\{{p^{(5)}}\right\\}-\dots\;$
(2.2)

where the plus and minus signs indicate set union and set difference rather
than arithmetic operations on the elements of the sequences. The sets
$\left\\{{p^{(n)}}\right\\}$ in 2.2 are defined as

$\left\\{{p^{(1)}}\right\\}={\left\\{{p_{n}}\right\\}}_{n=1}^{\infty}=\left\\{2,3,5,7,11,13,17,19,23,29,31,37,41,43,\dots\right\\}$
(2.3)

$=$ the set of all prime numbers, or what we will call the set of all first-
order primes ${P}^{(1)}$([3])

$\left\\{{p^{(2)}}\right\\}={\left\\{{p_{p_{n}}}\right\\}}_{n=1}^{\infty}=\left\\{3,5,11,17,31,41,59,67,83,109,127,157,\dots\right\\}$
(2.4)

$=$ the set of prime numbers with prime indexes, or second-order primes
${P}^{(2)}$ ([4])

$\left\\{{p^{(3)}}\right\\}={\left\\{{p_{p_{p_{n}}}}\right\\}}_{n=1}^{\infty}=\left\\{5,11,31,59,127,179,277,331,431,599,\dots\right\\}$
(2.5)

$=$ the set of second-order prime numbers with prime indexes, or third-order
primes ${P}^{(3)}$ ([5])

$\left\\{{p^{(4)}}\right\\}={\left\\{{p_{p_{p_{p_{n}}}}}\right\\}}_{n=1}^{\infty}=\left\\{11,31,127,277,709,\dots\right\\}$
(2.6)

$=$ the set of third-order prime numbers with prime indexes, or fourth-order
primes ${P}^{(4)}$ ([6])

$\left\\{{p^{(5)}}\right\\}={\left\\{{p_{p_{p_{p_{p_{n}}}}}}\right\\}}_{n=1}^{\infty}=\left\\{31,127,709,\dots\right\\}$
(2.7)

$=$ the set of fourth-order prime numbers with prime indexes, or fifth-order
primes ${P}^{(5)}$ ([7]) and so on and so forth.

To facilitate the construction of the unique subset of prime numbers
$\mathbb{P^{{}^{\prime}}}$, one can circumvent the bulk alternating summation
of the aforementioned prime number subsequences of increasing order by
arranging those subsequences side-by-side and summing laterally across the
rows to create the new $\mathbb{P^{{}^{\prime}}}$ sequence term-by-term as
follows:

(row) | $+p^{(1)}$ | $-p^{(2)}$ | $+p^{(3)}$ | $-p^{(4)}$ | $+p^{(5)}$ | $-p^{(6)}$ | $\dots$ | $p{{}^{\prime}}$
---|---|---|---|---|---|---|---|---
(1) | 2 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 2
(2) | 3 | 3 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 0
(3) | 5 | 5 | 5 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 5
(4) | 7 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 7
(5) | 11 | 11 | 11 | 11 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 0
(6) | 13 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 13
(7) | 17 | 17 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 0
(8) | 19 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 19
(9) | 23 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 23
(10) | 29 | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | $\longrightarrow$ | 29
(11) | 31 | 31 | 31 | 31 | 31 | $\longrightarrow$ | $\longrightarrow$ | 31
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
Table 1. Calculation of $\mathbb{P}^{\prime}$ as an alternating sum.

The infinite prime number sequence $\mathbb{P^{{}^{\prime}}}$ emerging in the
rightmost column of Table 1 is

$\mathbb{P^{{}^{\prime}}}=\left\\{{p{{}^{\prime}}}\right\\}=\left\\{2,5,7,13,19,23,29,31,37,43,47,53,59,61,71,\dots\right\\}.$

This is the new OEIS sequence A333242 ([8]). When $\mathbb{P^{{}^{\prime}}}$
is generated by the cross-summation method demonstrated in Table 1 above, this
sequence is represented by the equation:

$\displaystyle\phantom{{}+{}}\\{p_{1}^{(1)}-p_{1}^{(2)}+p_{1}^{(3)}-p_{1}^{(4)}+\cdots\\}+\\{p_{2}^{(1)}-p_{2}^{(2)}+p_{2}^{(3)}-p_{2}^{(4)}+\cdots\\}$
$\displaystyle+\\{p_{3}^{(1)}-p_{3}^{(2)}+p_{3}^{(3)}-p_{3}^{(4)}+\cdots\\}+\\{p_{4}^{(1)}-p_{4}^{(2)}+p_{4}^{(3)}-p_{4}^{(4)}+\cdots\\}+\cdots$
$\displaystyle+\\{p_{\infty}^{(1)}-p_{\infty}^{(2)}+p_{\infty}^{(3)}-p_{\infty}^{(4)}+\cdots\\}={\left\\{\displaystyle\sum\limits_{n=1}^{\infty}(-1)^{n-1}p_{r}^{(n)}\right\\}}_{r=1}^{\infty}$

where the superscript $(n)$ denotes the order of primeness as defined in Eqs.
2.3–2.7 and the subscript $r$ denotes the row number corresponding to the
natural number in the leftmost column of Table 1. Therefore, we find that the
new prime number sequence $\mathbb{P{{}^{\prime}}}$ can either be generated
via the bulk alternating summation of infinite prime number subsequences of
increasing order or it can be constructed finitely term-by-term by summing
laterally across the prime number subsequences of increasing order as
illustrated in Table 1 so that we have

$\mathbb{P^{{}^{\prime}}}={\left\\{{(-1)^{n-1}}\left\\{{p^{(n)}}\right\\}\right\\}_{n=1}^{\infty}}={\left\\{\displaystyle\sum\limits_{n=1}^{\infty}(-1)^{n-1}p_{r}^{(n)}\right\\}_{r=1}^{\infty}}.$

Recall that we previously defined $\mathbb{P^{{}^{\prime\prime}}}$ as the
complement of $\mathbb{P^{{}^{\prime}}}$ with respect to the set of all prime
numbers $\mathbb{P}$. As such, we have

$\mathbb{P^{{}^{\prime\prime}}}=\overline{\mathbb{P^{{}^{\prime}}}}=\mathbb{P{}}-\mathbb{P^{{}^{\prime}}}=\left\\{3,11,17,41,67,83,109,127,\dots\right\\}.$
(2.8)

The right-hand side of Eq. 2.8 represents the OEIS sequence A262275 ([9]). We
observe that $\mathbb{P^{{}^{\prime\prime}}}$ can also be defined in terms of
the following alternating sum of prime number subsequences of increasing order
by adjusting the starting index and reversing the sign of the terms on the
right hand side of Eq. 2.1:

$\mathbb{P^{{}^{\prime\prime}}}={\left\\{{(-1)^{n}}\left\\{{p^{(n)}}\right\\}\right\\}}_{n=2}^{\infty}\;=\left\\{{p^{(2)}}\right\\}-\left\\{{p^{(3)}}\right\\}+\left\\{{p^{(4)}}\right\\}-\cdots\;.$

Similarly, we can define even higher classes of prime number sequences formed
by the alternating sums of prime number subsequences of increasing order. If
we let

$\displaystyle\mathbb{P}^{(1)}$ $\displaystyle=\mathbb{P^{{}^{\prime}}}$
$\displaystyle\mathbb{P}^{(2)}$ $\displaystyle=\mathbb{P^{{}^{\prime\prime}}}$
$\displaystyle\mathbb{P}^{(3)}$
$\displaystyle=\mathbb{P^{{}^{\prime\prime\prime}}}$ $\displaystyle\vdots$

then

$\displaystyle\mathbb{P}^{(1)}$
$\displaystyle=\left\\{{(-1)^{n-1}}\left\\{{p^{(n)}}\right\\}\right\\}_{n=1}^{\infty}$
$\displaystyle\mathbb{P}^{(2)}$
$\displaystyle=\left\\{{(-1)^{n}}\left\\{{p^{(n)}}\right\\}\right\\}_{n=2}^{\infty}$
$\displaystyle\mathbb{P}^{(3)}$
$\displaystyle=\left\\{{(-1)^{n-1}}\left\\{{p^{(n)}}\right\\}\right\\}_{n=3}^{\infty}.$
$\displaystyle\vdots$

We conclude this section with a proof of our initial claim that
$\mathbb{P{{}^{\prime}}}$ is equivalent to the alternating sum of all prime
number sequences of increasing order beginning at $\mathbb{P}=P^{(1)}$.

######  Theorem 2.1.

$\mathbb{P^{{}^{\prime}}}=\left\\{{p^{(1)}}\right\\}-\left\\{{p^{(2)}}\right\\}+\left\\{{p^{(3)}}\right\\}-\left\\{{p^{(4)}}\right\\}+\cdots\;.$

###### Proof.

Theorem 1.1 states that

$\mathbb{P{}}=\mathbb{P{{}^{\prime}}}+\mathbb{P{{}^{\prime\prime}}}.$

Since $\mathbb{P{{}^{\prime}}}$ has been defined as the alternating sum
beginning at $P^{(1)}$, and $\mathbb{P{{}^{\prime\prime}}}$ has been defined
as the alternating sum beginning at $P^{(2)}$, then we have

$\mathbb{P^{{}^{\prime}}}=\left\\{{p^{(1)}}\right\\}-\left\\{{p^{(2)}}\right\\}+\left\\{{p^{(3)}}\right\\}-\left\\{{p^{(4)}}\right\\}+\cdots$

and

$\mathbb{P^{{}^{\prime\prime}}}=\left\\{{p^{(2)}}\right\\}-\left\\{{p^{(3)}}\right\\}+\left\\{{p^{(4)}}\right\\}-\left\\{{p^{(5)}}\right\\}+\cdots$

so that

$\displaystyle\mathbb{P{}}$
$\displaystyle=\mathbb{P{{}^{\prime}}}+\mathbb{P{{}^{\prime\prime}}}$
$\displaystyle=\left\\{{P^{(1)}}-{P^{(2)}}+{P^{(3)}}-{P^{(4)}}+{P^{(5)}}-\cdots\right\\}$
$\displaystyle+\left\\{{P^{(2)}}-{P^{(3)}}+{P^{(4)}}-{P^{(5)}}+{P^{(6)}}-\cdots\right\\}$
$\displaystyle=\mathbb{P{}}.\qed$

##  3\. N-Sieve

We discovered that the $\mathbb{P^{{}^{\prime}}}$ sequence can also be
constructed by a new sieving operation performed on the domain of natural
numbers $\mathbb{N}$ via the sequential selection and elimination of prime
number indexes on the natural number line. Hence, we introduce a new algorithm
which we will call the “N-sieve” for performing such an operation. The sieve
is accomplished as follows: Starting with $n=1$, choose the prime number with
subscript 1 (i.e., $p_{1}=2$) as the first term of the
$\mathbb{P^{{}^{\prime}}}$ sequence and eliminate that prime number from the
natural number line. We mark eliminated items by circling them. Then, proceed
forward on $\mathbb{N}$ from $1$ to the next available natural number. Since
$2$ was eliminated from the natural number line in the previous step, one
moves forward to the next available natural number that has not been
eliminated, which is $3$. $3$ then becomes the subscript for the next
$\mathbb{P^{{}^{\prime}}}$ term which is $p_{3}=5$, and $5$ is then eliminated
from the natural number line, and so on and so forth. Such a sieving operation
has been carried out in Table 2 for the natural numbers $1$ to $100$ and
should be easy to follow.

$\begin{array}[]{w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}w{c}{6pt}}1&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize2}⃝}&3&4&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize5}⃝}&6&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize7}⃝}&8&9&10&11&12&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize13}⃝}&14&15&16&17&18&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize19}⃝}&20\\\
21&22&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize23}⃝}&24&25&26&27&28&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize29}⃝}&30&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize31}⃝}&32&33&34&35&36&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize37}⃝}&38&39&40\\\
41&42&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize43}⃝}&44&45&46&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize47}⃝}&48&49&50&51&52&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize53}⃝}&54&55&56&57&58&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize59}⃝}&60\\\
\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize61}⃝}&62&63&64&65&66&67&68&69&70&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize71}⃝}&72&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize73}⃝}&74&75&76&77&78&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize79}⃝}&80\\\
81&82&83&84&85&86&87&88&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize89}⃝}&90&91&92&93&94&95&96&\raisebox{-2.5pt}{\huge\raisebox{2pt}{\normalsize97}⃝}&98&99&100\end{array}$
Table 2. N-sieve applied to first 100 integers

We will now designate $\mathbb{P{{}^{\prime}}}$, which has been alternately
created via the sieving operation above, by the new notation
$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{N}\end{math}}}}}\right\rfloor$
to indicate that the natural numbers have been sieved to produce this prime
number sequence. In other words,

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{N}\end{math}}}}}\right\rfloor=\mathbb{P{{}^{\prime}}}=\left\\{{2,5,7,13,19,23,29,31,37,43,47,53,59,61,71,\dots}\right\\}.$

Thus, we can define the N-sieve of the natural numbers in the terms that were
previously-derived for $\mathbb{P{{}^{\prime}}}$:

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{N}\end{math}}}}}\right\rfloor=\mathbb{P{{}^{\prime}}}={\left\\{{(-1)^{n-1}}\left\\{{p^{(n)}}\right\\}\right\\}}_{n=1}^{\infty}={{\left\\{\displaystyle\sum\limits_{n=1}^{\infty}(-1)^{n-1}p_{r}^{(n)}\right\\}}_{r=1}^{\infty}}.$

At this point, we conclude that the prime number sequence
$\mathbb{P{{}^{\prime}}}$ can be constructed in either of three different
ways:

  1. (1)

Via the bulk alternating summation of the infinite prime number subsequences
of increasing order;

  2. (2)

By arranging the prime number subsequences of increasing order side-by-side
and taking sums laterally across the subsequences row-by-row to create the
$\mathbb{P{{}^{\prime}}}$ sequence iteratively term-by-term as illustrated in
Table 1;

  3. (3)

By the new N-sieve introduced above.

##  4\. Extension of the Application

An interesting property that we observed earlier in the relationship between
the set of all prime numbers $\mathbb{P{}}$ and the complement prime number
sets is:

$\mathbb{P{}}-{I}=\mathbb{P}_{I},$

or more clearly,

$\mathbb{P{}}-\left\\{2,5,7,13,19,23,29,31,\dots\right\\}=\left\\{{p_{{}_{{}_{{2}}}},p_{{}_{{}_{{5}}}},p_{{}_{{}_{{7}}}},p_{{}_{{}_{{13}}}},p_{{}_{{}_{{19}}}},p_{{}_{{}_{{23}}}},p_{{}_{{}_{{29}}}},p_{{}_{{}_{{31}}}},\dots}\right\\}$

where the prime numbers of the sequence ${I}$ form the indexes for the
complement set of primes $\mathbb{P}_{{I}}$ such that

$\mathbb{P}_{{{I}}}=\\{p_{k}\mid k\in{I}\\}.$

When we take the new sieving operation to the next level by sifting the set of
all prime numbers $\mathbb{P{}}=P^{(1)}$, we produce the OEIS sequence A262275
([9]):

2 \raisebox{2pt}{\normalsize3}⃝ 5 7 \raisebox{2pt}{\normalsize11}⃝ 13
\raisebox{2pt}{\normalsize17}⃝ 19 23 29 31 37 \raisebox{2pt}{\normalsize41}⃝
43 47 53 59 61 \raisebox{2pt}{\normalsize67}⃝ 73 ….

We will now designate $\mathbb{P{{}^{\prime\prime}}}$, which has been
alternately created via the sieving operation above, by the notation
$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{P}\end{math}}}}}\right\rfloor$
to indicate that the prime numbers have been N-sieved to produce the prime
number sequence

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{P}\end{math}}}}}\right\rfloor=\mathbb{P{{}^{\prime\prime}}}=\mathbb{P}_{{I}}=\left\\{{3,11,17,41,67,\dots}\right\\}.$

Thus, we can now define the prime number sequence obtained by the N-sieving of
the prime numbers $\mathbb{P}$ as:

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{P}\end{math}}}}}\right\rfloor=\mathbb{P{{}^{\prime\prime}}}={\left\\{{(-1)^{n}}\left\\{{p^{(n)}}\right\\}\right\\}}_{n=2}^{\infty}={{\left\\{\displaystyle\sum\limits_{n=2}^{\infty}(-1)^{n}p_{r}^{(n)}\right\\}}_{r=1}^{\infty}}$

so that

$\mathbb{P{{}^{\prime}}}=\mathbb{P{}}-\mathbb{P{{}^{\prime\prime}}}\;\;\;\;\;\Rightarrow\;\;\;\;\;\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{N}\end{math}}}}}\right\rfloor=\mathbb{P{}}-\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{P}\end{math}}}}}\right\rfloor.$

As we continue to sieve prime number subsequences of increasing order, we
discover ([10], [11]):

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,{P}^{(2)}\end{math}}}}}\right\rfloor=$3
\raisebox{2pt}{\normalsize5}⃝ 11 17 \raisebox{2pt}{\normalsize31}⃝ 41
\raisebox{2pt}{\normalsize59}⃝ 67 83 109 127 157
\raisebox{2pt}{\normalsize179}⃝ 191…

and

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,{P}^{(3)}\end{math}}}}}\right\rfloor=$5
\raisebox{2pt}{\normalsize11}⃝ 31 59 \raisebox{2pt}{\normalsize127}⃝ 179
\raisebox{2pt}{\normalsize277}⃝ 331 431 599 709 919
\raisebox{2pt}{\normalsize1063}⃝ 1153…

which reveals the following relationships between sieved prime number
sequences of higher order:

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,{P^{(2)}}\end{math}}}}}\right\rfloor=\mathbb{P}^{(3)}=\mathbb{P_{\mathbb{P_{{I}}}}}$
(4.1)

and

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,{P^{(3)}}\end{math}}}}}\right\rfloor=\mathbb{P}^{(4)}=\mathbb{P_{\mathbb{P_{\mathbb{P_{{I}}}}}}}.$
(4.2)

The sieving operations on the prime number subsequences of higher order in Eq.
4.1–4.2 can be further defined as

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,{P^{(2)}}\end{math}}}}}\right\rfloor={P^{(2)}}-\mathbb{P}^{(2)}={P^{(2)}}-\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,{P^{(1)}}\end{math}}}}}\right\rfloor$

and

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,{P^{(3)}}\end{math}}}}}\right\rfloor={P^{(3)}}-\mathbb{P}^{(3)}={P^{(3)}}-\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,P^{(2)}\end{math}}}}}\right\rfloor.$

With these new tools, we can now define all higher-order prime number
subsequences of the form (see [2])

$\left\\{{p_{p_{._{._{._{p_{p_{n}}}}}}}}\right\\}{\Biggr{\rvert}_{n=1}^{\infty}}\\\
$

in terms of the sieved prime number sequences as follows:

$\displaystyle\left\\{{p_{n}}\right\\}{\Biggr{\rvert}_{n=1}^{\infty}}$
$\displaystyle=P^{(1)}=\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{N}\end{math}}}}}\right\rfloor+\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,P^{(1)}\end{math}}}}}\right\rfloor=\mathbb{P}^{(1)}+\mathbb{P}^{(2)}$
$\displaystyle\left\\{{p_{p_{n}}}\right\\}{\Biggr{\rvert}_{n=1}^{\infty}}$
$\displaystyle=P^{(2)}=\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,P^{(1)}\end{math}}}}}\right\rfloor+\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,P^{(2)}\end{math}}}}}\right\rfloor=\mathbb{P}^{(2)}+\mathbb{P}^{(3)}$
$\displaystyle\left\\{{p_{p_{p_{n}}}}\right\\}{\Biggr{\rvert}_{n=1}^{\infty}}$
$\displaystyle=P^{(3)}=\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,P^{(2)}\end{math}}}}}\right\rfloor+\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,P^{(3)}\end{math}}}}}\right\rfloor=\mathbb{P}^{(3)}+\mathbb{P}^{(4)}$
$\displaystyle\vdots$

##  5\. Venn Diagram Analysis

Figure 1. Venn diagram relating the various sequences.

Figure 1 is a Venn diagram which illustrates the interrelationships between
the prime number subsequences of higher order and the N-sieved prime number
sequences heretofore disclosed.

We interpret the diagram as follows: The outermost yellow ring $\mathbb{C}$ is
the composite numbers. Starting with the outermost blue ring, we see that the
range of prime numbers $P^{(1)}-P^{(2)}$ is a subset of $\mathbb{P}^{(1)}$ as
denoted by $\subset\mathbb{P}^{(1)}$ inscribed inside the outer blue ring.
Continuing on to the next blue ring inside the outer blue ring, we see that
the range of prime numbers $P^{(3)}-P^{(4)}$ constitute the next subset of
prime numbers to include in $\mathbb{P}^{(1)}$ denoted by
$\subset\mathbb{P}^{(1)}$ inscribed inside that blue ring. But the second
inner blue ring in the Venn diagram not only contains the next subset of prime
numbers to be included in the range of $\mathbb{P}^{(1)}$, but it also
contains the first set of primes to be included in the next higher sieved set
$\mathbb{P}^{(3)}$. This pattern continues to infinity as each inner blue ring
continues to contribute additional primes to include in $\mathbb{P}^{(1)}$ in
addition to providing the beginning terms of a new sequence of a higher sieved
set of primes which in turn will also continue to gather new prime numbers for
its sequence as each blue ring is encountered all the way to infinity.

This process works similarly for the alternate sets of sieved primes as
follows: Starting with the outermost red ring, we see that the range of prime
numbers $P^{(2)}-P^{(3)}$ is a subset of $\mathbb{P}^{(2)}$ as denoted by
$\mathbb{P}^{(2)}$ inscribed inside the outer red ring. Continuing on to the
next red ring inside the outer red ring, we see that the range of prime
numbers $P^{(4)}-P^{(5)}$ constitute the next subset of prime numbers to
include in $\mathbb{P}^{(2)}$ denoted by $\subset\mathbb{P}^{(2)}$ inscribed
inside that inner red ring. But the second inner red ring not only contains
the next subset of prime numbers to be included in the range of
$\mathbb{P}^{(2)}$, but it also contains the first set of primes to be
included in the next higher sieved set $\mathbb{P}^{(4)}$. This pattern
continues to infinity as each inner red ring continues to contribute
additional primes to be included in $\mathbb{P}^{(2)}$ in addition to
providing the beginning terms of a new sequence of a higher sieved set of
primes which in turn will also continue to gather new prime numbers for its
sequence as each inner red ring is encountered all the way to infinity.

In order to add more clarity to the Venn diagram, we define the elements of
the first six annuli as follows:

Ring No. 1: $\mathbb{N}-\mathbb{P}=\mathbb{C}$

Ring No. 2: $P^{(1)}-P^{(2)}\subset\mathbb{P}^{(1)}$

Ring No. 3: $P^{(2)}-P^{(3)}\subset\mathbb{P}^{(2)}$

Ring No. 4: $P^{(3)}-P^{(4)}\subset\mathbb{P}^{(3)}\subset\mathbb{P}^{(1)}$

Ring No. 5: $P^{(4)}-P^{(5)}\subset\mathbb{P}^{(4)}\subset\mathbb{P}^{(2)}$

Ring No. 6:
$P^{(5)}-P^{(6)}\subset\mathbb{P}^{(5)}\subset\mathbb{P}^{(3)}\subset\mathbb{P}^{(1)}$

##  6\. General Case

At this point, we present a theorem for the general case of Theorem 1.1:

######  Theorem 6.1.

A unique subset of prime numbers $\mathbb{P}^{(i)}$ exists which, when applied
as indexes to a subset of prime numbers $P^{(i)}$, yields the complement set
of primes $\mathbb{P}^{(i+1)}$ so that

$\mathbb{P}^{(i)}=P^{(i)}-\mathbb{P}^{(i+1)}.$ (6.1)

We will show that if there is a subset $\mathbb{P}^{(i)}\subset P^{(i)}$ such
that $\mathbb{P}^{(i)}$ and $\mathbb{P}^{(i+1)}$ are a partition of $P^{(i)}$,
that is if

$P^{(i)}=\mathbb{P}^{(i)}\cup\mathbb{P}^{(i+1)}$

and

$\mathbb{P}^{(i)}\cap\mathbb{P}^{(i+1)}=0,$

then $\mathbb{P}^{(i)}$ is unique and $\mathbb{P}^{(i)}$ and
$\mathbb{P}^{(i+1)}$ are complement sets of $P^{(i)}$.

###### Proof.

We begin by showing that $\mathbb{P}^{(i)}$ of Eq. 6.1 is unique and that the
first $n$ elements of $\mathbb{P}^{(i)}$ and $\mathbb{P}^{(i+1)}$ determine
the $n+1st$ element of $\mathbb{P}^{(i)}$. First, we write

$q_{1},\dots,q_{n}$

to designate the smallest elements of $\mathbb{P}^{(i)}$ in increasing order.
And because the $nth$ prime function $p_{n}$ is a monotonic increasing
function, it follows that

$p_{q_{1}},\dots,p_{q_{n}}$

are the smallest elements of $\mathbb{P}^{(i+1)}$ in corresponding increasing
order. Now let $q$ be the smallest prime number that is different from all
$q_{1},\dots,q_{n}$ and $p_{q_{1}},\dots,p_{q_{n}}$. We know that either
$q\in\mathbb{P}^{(i)}$ or $q\in\mathbb{P}^{(i+1)}$, and our claim is that
$q\in\mathbb{P}^{(i)}$. But if we suppose to the contrary that
$q\in\mathbb{P}^{(i+1)}$, then there must be a prime number $r$ such that
$q=p_{r}$. Now we assumed that $q>p_{q_{n}}$, and it follows from the
definition that $r>q_{n}$ and $r<p_{r}=q$. But this means that $r$ satisfies
the same conditions as $q$: i.e., that $r$ is the smallest prime different
from all $q_{1},\dots,q_{n}$ and $p_{q_{1}},\dots,p_{q_{n}}$. So we should
have chosen $r$ instead of $q$ because $r$ is smaller than the $q$ we assumed
to be an element of $\mathbb{P}^{(i+1)}$. Therefore, our assumption that
$q\in\mathbb{P}^{(i+1)}$ is a contradiction, and the smallest prime $q$
different from all $q_{1}$,…,$q_{n}$ and $p_{q_{1}}$,…,$p_{q_{n}}$ must be an
element of $\mathbb{P}^{(i)}$ because the elements of $\mathbb{P}^{(i)}$,
which serve as the subscripts of $\mathbb{P}^{(i+1)}$, always lag behind the
elements of $\mathbb{P}^{(i+1)}$. Furthermore, since all prime numbers
$P^{(i)}$ are consumed in the process of adding the smallest available prime
number different from all $q_{1},\dots,q_{n}$ and $p_{q_{1}},\dots,p_{q_{n}}$
to $\mathbb{P}^{(i)}$, then we have shown that $\mathbb{P}^{(i)}$ and
$\mathbb{P}^{(i+1)}$ are complement sets of $P^{(i)}$. Thus, we have shown by
induction on $i$ that Theorem 6.1, as the general case of Theorem 1.1, holds
true. ∎

We note here that the prime number sequences

$\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{\mathbb{P^{{}^{\prime}}}}\end{math}}}}}\right\rfloor,\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{\mathbb{P^{{}^{\prime\prime}}}}\end{math}}}}}\right\rfloor,\left\lfloor{\raisebox{-0.3pt}{\\!{\dashuline{\begin{math}\,\mathbb{\mathbb{P^{{}^{\prime\prime\prime}}}}\end{math}}}}}\right\rfloor,\dots$

do not exist because the complement set of primes corresponding to the indexes
of the prime numbers in each of these sequences has already been sifted out
via the N-sieving operation. In other words, one cannot sift any terms out of
a prime number sequence which has already been N-sieved. As an example,
consider the prime number subsequence

$\mathbb{P^{{}^{\prime}}}=\left\\{{p{{}^{\prime}}}\right\\}=\left\\{2,5,7,13,19,23,29,31,37,43,47,53,59,61,71,\dots\right\\}.$

If we attempt to apply the first term in this sequence as an index to the set
of prime numbers for the purpose of eliminating that term from
$\mathbb{P^{{}^{\prime}}}$, we see that $p_{2}=3$ isn’t present in
$\mathbb{P^{{}^{\prime}}}$. Neither is the next term $p_{5}=11$. And thus is
the case for every term in $\mathbb{P^{{}^{\prime}}}$ for which one would
attempt to apply as an index to the set of all prime numbers for the purpose
of eliminating primes from $\mathbb{P^{{}^{\prime}}}$.

##  7\. Conclusion

In this paper, we derived some of the basic properties of alternating sums of
sequential sets of higher-order prime number subsequences and presented
several methods available to derive them. It is the author’s hope that this
research will shed some light on new ways to explore prime number sequences of
higher order and that the methods presented herein will contribute new ideas
to that end.

##  8\. Acknowledgements

We gratefully acknowledge and thank the Number Theory Editor and the referees
of the Missouri Journal of Mathematical Sciences for their helpful comments
during the editing process, and we particularly express our gratitude to them
for their help with the outlining of the proofs of the theorems.

## References

  * [1] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
http://oeis.org/wiki/Higher-order prime numbers.

  * [2] Kevin A Broughan and A. Ross Barnett. On the Subsequence of Primes Having Prime Subscripts. Journal of Integer Sequences, Vol. 12 (2009) Article 09.2.3.
  * [3] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A000040 by N. J. A. Sloane.

  * [4] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A006450 by Jeffrey Shallit.

  * [5] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A038580 by Brian Galebach.

  * [6] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A049090 by Neil Fernandez.

  * [7] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A049203 by Neil Fernandez.

  * [8] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A333242 by Michael P. May, March 2020.

  * [9] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A262275 by Zak Seidov and Robert G. Wilson.

  * [10] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A333243 by Michael P. May, March 2020.

  * [11] N. J. A. Sloane. Online Encyclopedia of Integer Sequences.
Sequence A333244 by Michael P. May, March 2020.