# The Properties and Relations of Prime Numbers

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```					The Properties and Relations of Prime Numbers

Brendan Flanagan-Rosario

Evanston Township High School

January 25, 2010
Flanagan-Rosario, Brendan

Introduction

A prime number, or a positive integer p>1 that has no positive integer divisors other than

1 and p itself [1], is probably the most important type of integer. A solid understanding

of prime numbers considerably impacts all of mathematics. Specific “real world”

applications would primarily be in the field of cryptography, such as creating new

encryption systems or cracking old ones. Mathematicians have spent centuries analyzing

primes, but much remains unknown.

The overall purpose of this project was to discover new information about primes.

Precisely, the goals of this project were to:

   Investigate the Goldbach Conjecture, which states that all positive even integers ≥

2 can be expressed as the sum of two (not necessarily distinct) primes [2].

   Create a convenient (ideally closed-form) method that relates primes [3].

I took many different approaches when solving these problems, ranging from

coordinate geometry to matrix theory and set theory, despite that these problems are

primarily of curiosity within number theory.

This paper examines the following results that were found:

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Flanagan-Rosario, Brendan

   New various methods that can be used to prove the Goldbach Conjecture and their

faults

   Implications of a chart that relates the factors of positive integers; mainly,

algorithm derived from this chart that produces prime numbers and behaves

similarly to the “next prime” function [4].

Methods

Following Prime (p≥n) Algorithm

I used many methods to finding any sort of algorithm that relates primes – making clever

guesses and testing them on my calculator, drawing and then analyzing charts and

diagrams, combining existing functions, etc. One of these methods actually led to

significant findings.

At first, I thought of arranging the primes in the chart shown below could be useful:

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Flanagan-Rosario, Brendan

1 1 1 1 …

2      2        2       2       2       2 …

3          3            3           3      …

5                   5                   5 …

7                           7              …

11                                          11 …

13                                               …

…

Note how each 2 in a row occurs 2 spaces after the previous 2, each 3 in a row occurs 3

spaces after the previous 3 in a row, etc.

While the chart may help to visualize the distribution of primes, I was not able to extract

any useful relations from the chart. So, I decided to make a similar chart that included all

of the positive integers. A section of the chart is shown:

x     I n c r e a s e s

y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Flanagan-Rosario, Brendan

2   2       2       2       2        2         2         2         2   2    2        2    2        2        2    2

I    3       3           3           3              3              3        3        3         3            3         3

n 4              4               4                  4                   4        4             4                 4

c    5               5                    5                        5             5                 5                  5

r    6                   6                          6                       6                  6                      6

e    7                       7                                7                      7                           7

a    8                           8                                      8                      8

s    9                               9                                      9                               9

e 10                                     10                                     10                                   10

s 11                                          11                                         11

12                                             12                                         12

13                                                  13                                             13

14                                                       14                                                 14

15                                                            15                                                 15

16                                                                 16

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Flanagan-Rosario, Brendan

17                                              17

18                                                   18

19                                                        19

20                                                             20

21                                                                  21

22                                                                       22

23                                                                            23

24                                                                                 24

One fascinating and useful pattern I discovered from this chart is how one can “play

pinball” with the chart by “bouncing” off of numbers and converging to a prime-

numbered column (a column where the bolded number along the 45-degree diagonal is

prime) whose x-coordinate is at least that of the original column. In other words, this can

be used to generate a prime number greater than or equal to a given integer n, such that n

≥ 2. Here are the details of how this algorithm works (the "rules of this game"):

1. Choose a number/location along the bolded diagonal

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Flanagan-Rosario, Brendan

2. Repeat the following indented steps (2-A, 2-B-i, 2-B-ii) until the space vertically "in

front" is 1 (the y coordinate is 2)

2-A. Move forward until a number is encountered. The current location is the

space right before this number (1 y coordinate greater).

2-B-i. If there is a number in the space directly to the right (1 x coordinate

greater), then move left until a number is encountered. The current location is the space

to the right (1 x coordinate greater) of this space containing the number. Go back to step

2.

2-B-ii. If there is not a number in the space directly to the right, then move right

until a number is encountered. The current location is the space directly to the left of this

space containing the number (1 x coordinate less). Go back to step 2.

3. When y = 2, then x = the final result (x coordinates for each column can be found by

looking at the bolded and italicized diagonal)

It can be translated into mathematical pseudo-code like so (with the floor() function

returning the largest integer less than or equal to the parameter it accepts [5]):

let x = y = the input number (n)

start loop

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Flanagan-Rosario, Brendan

y=(the largest factor of x that is less than y)+1

if y≠2, continue to the next step; otherwise, x equals the final result

if x + 1 is divisible by y, then x=y*floor(x/y)+1;

otherwise, x=y*(floor(x/y)+1)-1

repeat loop

A complete C++ implementation following the structure of the pseudo-code:

#include <iostream>
#include <cmath>

using namespace std;

main()
{
int x,y,t;
cout << "Number:";
cin >> x; // Store the input number in x and y
y=x;

for(;;){ // Repeat steps (loop-exiting condition occurs later in the algorithm)
for(t=y-1;t>0;t--){ // Find the largest factor of x that is less than y, and store 1 plus that
value in y

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Flanagan-Rosario, Brendan

if(!(x % t))break;
}
y=t+1;
if(y==2)break; // If y = 2, then x = the final result, so the algorithm concludes
if(!((x + 1) % y)) x=y*(int)(x/y)+1; // If there is a number directly to the right, then move
left
else
x=y*((int)(x/y) + 1)-1; // Otherwise, move right
}
cout << "Result:" << x;
return 0;
}
The algorithm is simpler than it may seem. The following diagram traces the path using

the algorithm with an initial number of 8 (the “#” symbol represents the location within

the chart of the course of the algorithm):

1 1 1 1 1 1 1 1 1 1                 1     1      1

2       2       2       2   2       2 #          2

3           3           3       3 #      #       3

4               4           4 # #                4

5                   5       # #     5

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Flanagan-Rosario, Brendan

6                    6                          6

7                        7

8                            8

9                                9

10                                    10

11                                         11

12                                              12

…                                                    …

Progression of x and y values

x    y    Comment

x and y equal the initial number (on the bolded and italicized

8 8      diagonal)

8 5      y=(the largest factor of x that is less than y)+1

9 5      x=y*(floor(x/y)+1)-1

9 4      y=(the largest factor of x that is less than y)+1

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Flanagan-Rosario, Brendan

10 4       x=y*(floor(x/y)+1)-1

10 3       y=(the largest factor of x that is less than y)+1

11 3       x=y*(floor(x/y)+1)-1

11 2       y=(the largest factor of x that is less than y)+1

y=2 in the previous step, can't go any further; x=11 is the final

result

Here is another example, this time demonstrating how the x coordinate throughout the

algorithm can also decrease (using the relevant section of the chart):

1            1      1          1             1      1
2 #               2                    2
3                               3
4 #                   #                4
5                                                   5
6
7
#                                         #                8
9
10

12
13

18
19

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Flanagan-Rosario, Brendan

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35
36
37
38
39
40

I initially thought that it generated the smallest prime greater than or equal to a given

number, but it turns out that the prime that is generated (and is greater than or equal to a

given number) is not always the smallest (for example, an input of 48 produces 59, even

though 53 is also prime).

Goldbach Conjecture

As with the previous task, I approached the Goldbach Conjecture in many ways. I tried

proving two equivalent forms of the Goldbach Conjecture:
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Flanagan-Rosario, Brendan

   Every positive integer ≥ 2 can be expressed as the average of two primes

   For every integer m ≥ 2, there is some integer n such that m + n and m – n are

both primes.

My preliminary attempts followed this core reasoning:

Let m be an integer such that m ≥ 2. Consider the set of all positive integers greater than

or equal to 2 and less than or equal to m that are relatively prime to m (two integers are

relatively prime if they do not share any positive divisors other than 1 [6]). Then let n

equal the smallest number that shares at least one factor with each of the numbers in this

set. Thus, the set of integers m – n, m – n2, …, m – nk for positive values of m minus n to

a power contains only prime numbers....

However, this reasoning was flawed, as sometimes n > m – 2, leading to non-prime

results. Later, I thought of two other approaches.

One approach was to try to represent numbers in a new way that would make the

Goldbach Conjecture more obvious. I decided that I could give numbers "dimensions"

and have them represented as vectors or matrices using their prime factorizations and

listing the smallest factors first.

For example, 495, which factors into 32*5*11 (and would have a dimension of 4 because

of the two 3s, 5 and 11 for 4 numbers total), could be expressed as <3, 3, 5, 11> or [[3 0 0

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Flanagan-Rosario, Brendan

0 ][0 3 0 0 ][0 0 5 0][0 0 0 11]] (in which case the determinant of this matrix would equal

the original number). All primes would be 1-dimensional.

To compare two numbers of different dimensions (say 6=2*3 verses 495), one could give

them the same dimensions like so: <1, 1, 2, 3> and <3, 3, 5, 11>.

These vectors or matrices could be represented on a set of axes, with the amount of axes

being determined by a number's dimension.

Each axis would only contain the primes, possibly preceded by 1 and maybe 0. Starting

at 0 would mean that the total "units" enclosed (in other words, the area for a 2-

dimensional number and the volume for a 3-dimensional number) would correspond to

the numbers' value.

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Flanagan-Rosario, Brendan

Or, instead of multiplying the values on each axis, they could be added, giving

"dimension" a different meaning.

This general approach leads to many more methods to try to prove the Goldbach

Conjecture. The primary method under this system that I tried was to see if I could show

that any n-dimensional number, for an integer n > 2, can be equivalently expressed as a

number of dimension n-1, assuming that dimensions are determined by adding numbers

and not multiplying them. I was unsuccessful, though. Although this approach could be

explored a lot more (especially by applying methods of linear algebra), I decided to turn

to a different approach. The dimension/vector/matrix idea may only turn out to be

mathematical nonsense.

Let m be an even integer such that m ≥ 2. It is relatively clear that every 2m can be

expressed as the sum of 2 numbers within the set {2, 3, 5, 7, ..., 2n + 1, …}. (4 = 2 + 2, 6

= 3 + 3, 8 = 3 + 5, 10 = 3 + 7, … }It is also not too difficult to show that every 2m can be

expressed as the sum of 2 numbers within another subset of the natural numbers (and

superset of the primes), although the cases get a bit messier. Why should this logic

suddenly stop working?

I initially tried to use induction with this method, but my work was getting convoluted.

What significantly helped was to redefine the problem in terms of sets.

First, some explanations of standard symbols and conventions:

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Flanagan-Rosario, Brendan

   Capital letters are used to denote sets, while lower-case letters are used to denote

integer variables

   x|y means x divides y; likewise, x∤y means that x does not divide y

   x ∈ S means x exists within the set S; likewise, x

   T ⊆ S means the set T is a subset of S, or that every element in T exists in S

   S C returns the complement of S (all elements that do not exist in S) within some

universal set, which is {2, 3, 4, 5, …} for the purposes of this paper

   T ⋂ S returns the set of all elements that exist in T and S

In addition, the operator ⊞ in this proof is defined in the following ways for sets S and T:

   S ⊞ T = the set resulting from all possible sums of one element from S and one

element from T; precedence in the order of operations is the same as the

precedence for the union operator ⋃ (this definition is not used in the rest of this

paper, but it still could be useful, so it is defined anyway)

   S⊞ = the set resulting from all possible sums of two elements (not necessarily

distinct) from S; precedence in the order of operations is the same as the

precedence for the complement operator c

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Flanagan-Rosario, Brendan

Let N denote the set of natural numbers and let T = 2*N + 2; e ∈ T. Let n, m, x, y, j, k

and all xi, yi, ki, ai, bi ∈ N + 1 and i ∈ N. a and b are prime, where b > a.

Note that the set of primes P = (2*N + 2)C ⋂ (3*N + 3)C ⋂ (4*N + 4)C ⋂ …

The objective is to prove T ⊆ P⊞.

Lemma 1: For a set S = a*N + a, T ⊆ ((a*N + a)C)⊞.

Proof:

Case a ≠ 3

    If e ≥ 6, then e = 3 + (e – 3), where 3 ∈ SC ;

e – 3 ∈ SC because e and 3 are coprime and e – 3 ≥ 3

    If e = 4, then e = 2 + 2, where 2 ∈ SC

Case a = 3

    If e ≥ 8, then e = 5 + (e – 5), where 5 ∈ (3*N+3)C ;

e – 3 ∈ (3*N+3)C because e and 5 are coprime and e – 5 ≥ 3

    If e = 6, then e = 3 + 3, where 3 ∈ (3*N+3)C

    If e = 4, then e = 2 + 2, where 2 ∈ (3*N+3)C

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Flanagan-Rosario, Brendan

Lemma 2: Let A = (a*N + a)C and B = (b*N + b)C . If e = x + y, where x ∈ A and y ∈ B,

then e = x2 + y2, where x2, y2 ∈ A and x2, y2 ∈ B (x2, y2 ∈ A ⋂ B). (Equivalently, if T ⊆

((a*N + a)C)⊞ and T ⊆ ((b*N + b)C)⊞, then T ⊆ ((a*N + a)C ⋂ (b*N + b)C)⊞ )

Proof:

Case 1: x ∈ A ⋂ BC, y ∈ A ⋂ B

e = b*k + y, where k≥2, a∤k and one of the following statements separated by a

semicolon is true (note e≥8 because b≥3, as b>a≥2, and y≥2): a,b∤y; y = a; y = b

1. If y = b and a∤e, then x2, y2 = e – a, a.

2. If [y = b and a|e] or [y ≠ b], then one of x2, y2 = e – b - 2, b + 2, x2, y2 = e – b - 3, b

+ 3, or x2, y2 = e – b – 4, b + 4 result in x2, y2 ∈ A ⋂ B

Case 2: x ∈ A ⋂ BC, y ∈ AC ⋂ B

e = b*k + a*j, where j, k≥2; a∤k, b∤j, and thus a,b∤e. By the logic of step 2 in case 1, x2,

y2 can be found.

Note: x and y are interchangeable, thus these are the only two cases.

//

From here, there are many directions the proof could follow. These simple lemmas have

set a foundation, showing that positive even integers greater than or equal to 4 can be

expressed as the sum of two elements from supersets of primes – supersets that are

closely linked to primes in that they exclude multiples of certain numbers. One might try

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Flanagan-Rosario, Brendan

to prove that if T ⊆ (a*N + a) C ⋂ ((b1*N + b1) C ⋂ (b2*N + b2) C ⋂ …) and T ⊆ (a*N +

a) C ⋂ ((c1*N + c1) C ⋂ (c2*N + c2) C ⋂ …), then T ⊆ (a*N + a) C ⋂ ((b1*N + b1) C ⋂

(b2*N + b2) C ⋂ …) ⋂ ((c1*N + c1) C ⋂ (c2*N + c2) C ⋂ …) (if this were proven, then this

fact could be repeatedly used to show that T ⊆ P⊞). I have not had any success with any

directions, but it is still useful to examine possible faulty proofs based off of this work

(the correct logic within the overall faulty proof could aid in developing a completely

correct proof, and knowing what logic to avoid is beneficial).

An example of a faulty proof based off of lemma 1:

Induction on the number of intersections of sets and the largest value n (as previously

defined) in the statement T ⊆ (2*N + 2)C ⋂ (3*N + 3)C ⋂ (4*N + 4)C ⋂ … ⋂ (n*N + n)C

(so there are n – 2 intersections):

Base case: T ⊆ (2*N + 2)C ; true by lemma 1s

Inductive step: Assume T ⊆ (2*N + 2)C ⋂ (3*N + 3)C ⋂ (4*N + 4)C ⋂ … (k*N + k)C; try

to prove T ⊆ (2*N + 2)C ⋂ (3*N + 3)C ⋂ (4*N + 4)C ⋂ … (k*N + k)C ⋂ ((k + 1)*N + k +

1)C

Because T ⊆ (2*N + 2)C ⋂ (3*N + 3)C ⋂ (4*N + 4)C ⋂ … (k*N + k)C by the inductive

hypothesis and T ⊆ ((k + 1)*N + k + 1)C by lemma 1, T ⊆ (2*N + 2)C ⋂ (3*N + 3)C ⋂

(4*N + 4)C ⋂ … (k*N + k)C ⋂ ((k + 1)*N + k + 1)C .

//

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Flanagan-Rosario, Brendan

Besides inducting in an unusual way on the number of intersections and the largest value

n (which, if it is incorrect, could most likely be revised), there is a critical flaw when

proving the inductive step.

The inductive hypothesis says e = s + t, for some s, t ∈ (2*N + 2)C ⋂ (3*N + 3)C ⋂ (4*N

+ 4)C ⋂ … ⋂ (k*N + k)C . Lemma 1 lets us says e = q + r, for some q, r ∈ ((k + 1)*N + k

+ 1)C. It is not necessarily known, however, if q, r ∈(2*N + 2)C ⋂ (3*N + 3)C ⋂ (4*N +

4)C ⋂ … ⋂ (k*N + k)C or if s, t ∈ ((k + 1)*N + k + 1)C, so one cannot conclude that T ⊆

(2*N + 2)C ⋂ (3*N + 3)C ⋂ (4*N + 4)C ⋂ … ⋂ (k*N + k)C ⋂ ((k + 1)*N + k + 1)C.

Conclusions and Future Studies

The section of this paper focusing on the Goldbach Conjecture serves as a guide to

anybody attempting to prove it or similar conjectures. A foundation for a proof of the

Goldbach Conjecture using set theory was produced, along with the idea of giving

numbers "dimensions" and treating them as vectors/matrices. The logic in this paper

suggests that the Goldbach Conjecture is in fact true; supersets of primes similar to the

set of primes itself were proven to work.

A unique algorithm for generating primes derived from a simple chart was produced.

This algorithm could possibly be modified to model the next prime function, which

would be extremely useful [4]. The reason why this method of "bouncing" off of

numbers to end in a prime-numbered column works could be further investigated and

may finally lead to a nice closed-form relation between primes. Finally, this algorithm

could be used in a primality test, an area of interest within cryptography.
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Flanagan-Rosario, Brendan

Acknowledgements

   Dr. Mark Vondracek, my physics teacher, who informed me about the Intel Talent

Search and helped me since I started working on my project, giving plenty of

feedback and guiding me through the research process

   Dr. Sara Quinn, a professor at Northwestern University who also is my math

teacher, for proofreading part of my work and teaching me more about number

theory during class

   Dr. Paul Sally Jr., a professor at the University of Chicago and Directory of

Undergraduate Studies, for giving me ideas and suggesting other professors

whom I could show my work to

   Brendan Fletcher, for looking over some of the Goldbach Conjecture information

Bibliography

[1] Weisstein, Eric W. "Prime Number." From MathWorld--A Wolfram Web Resource.

[2] Weisstein, Eric W. "Goldbach Conjecture." From MathWorld--A Wolfram Web

Resource. http://mathworld.wolfram.com/GoldbachConjecture.html

[3] Weisstein, Eric W. "Closed-Form Solution." From MathWorld--A Wolfram Web

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Flanagan-Rosario, Brendan

Resource. http://mathworld.wolfram.com/Closed-FormSolution.html

[4] Weisstein, Eric W. "Next Prime." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/NextPrime.html

[5] Weisstein, Eric W. "Floor Function." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/FloorFunction.html

[6] Weisstein, Eric W. "Relatively Prime." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/RelativelyPrime.html

All online sources are were viewed November 15, 2009

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Flanagan-Rosario, Brendan

Abstract
TITLE: The Properties and Relations of Prime Numbers
NAME: Brendan Flanagan-Rosario
SCHOOL: Evanston Township High School in Evanston, Illinois
SPONSOR: Evanston Township High School Teacher: Dr. Mark Vondracek

The purpose of this project was to:

   Investigate the Goldbach Conjecture, which states that all positive even integers ≥

2 can be expressed as the sum of two (not necessarily distinct) primes [2].

   Create a convenient (ideally closed-form) method that relates primes [3].

These tasks are tightly linked, as a solution to one may lead to a solution to the other.

Many different approaches were taken in solving these problems, ranging from

coordinate geometry to matrix theory and set theory, despite that these problems are

primarily of curiosity within number theory.

This paper examines these following results found:

   New various methods in proving the Goldbach Conjecture and their faults

   Implications of a chart that relates the factors of positive integers; mainly, an

algorithm derived from this chart that produces prime numbers and behaves

similarly to the “next prime” function [4]

22

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