On the Prime Number Subset of the Fibonacci Numbers Lacey Fish 1 - - PowerPoint PPT Presentation

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On the Prime Number Subset of the Fibonacci Numbers Lacey Fish 1 - - PowerPoint PPT Presentation

Jun 11, 2023 101 likes •1.07k views

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach On the Prime Number Subset of the Fibonacci Numbers Lacey Fish 1 Brandon Reid 2 Argen West 3 1 Department of Mathematics

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

On the Prime Number Subset of the Fibonacci Numbers

Lacey Fish1 Brandon Reid2 Argen West3

1Department of Mathematics

Louisiana State University Baton Rouge, LA

2Department of Mathematics

University of Alabama Tuscaloosa, AL

3Department of Mathematics

University of Louisiana Lafayette Lafayette, LA

SMILE Presentations, 2010

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

What is a sieve?

What is a sieve?

A sieve is a method to count or estimate the size of “sifted sets”

  • f integers. Well, what is a sifted set? A sifted set is made of

the remaining numbers after filtering.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

What is a sieve?

What is a sieve?

A sieve is a method to count or estimate the size of “sifted sets”

  • f integers. Well, what is a sifted set? A sifted set is made of

the remaining numbers after filtering.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

History

Two Famous and Useful Sieves

Sieve of Eratosthenes Brun’s Sieve

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

History

Two Famous and Useful Sieves

Sieve of Eratosthenes Brun’s Sieve

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

History

The Sieve of Eratosthenes

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Brun Sieve

The Brun sieve is a generalized method compared to the Eratosthenes sieve. It allows us to sieve any set A with a designated set P. It is formally stated as: S(A, P, z) = |A\

  • p∈P(z)

Ap|

But what does that mean???

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Brun Sieve

The Brun sieve is a generalized method compared to the Eratosthenes sieve. It allows us to sieve any set A with a designated set P. It is formally stated as: S(A, P, z) = |A\

  • p∈P(z)

Ap|

But what does that mean???

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

slide-45
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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Brun Sieve

The Brun sieve is a generalized method compared to the Eratosthenes sieve. It allows us to sieve any set A with a designated set P. It is formally stated as: S(A, P, z) = |A\

  • p∈P(z)

Ap|

But what does that mean???

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

For clarity, let us look at an example

We may take A = {5, 6, 10, 11, 12, 13, 18, 20, 22, 24, 28, 35} and P = {2, 7}. By sifting A with the given P, we see A2 = {6, 10, 12, 18, 20, 22, 24, 28}, and A7 = {28, 35} . We are left with S(A, P, z) = |{5, 11, 13}| = 3.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

For clarity, let us look at an example

We may take A = {5, 6, 10, 11, 12, 13, 18, 20, 22, 24, 28, 35} and P = {2, 7}. By sifting A with the given P, we see A2 = {6, 10, 12, 18, 20, 22, 24, 28}, and A7 = {28, 35} . We are left with S(A, P, z) = |{5, 11, 13}| = 3.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

For clarity, let us look at an example

We may take A = {5, 6, 10, 11, 12, 13, 18, 20, 22, 24, 28, 35} and P = {2, 7}. By sifting A with the given P, we see A2 = {6, 10, 12, 18, 20, 22, 24, 28}, and A7 = {28, 35} . We are left with S(A, P, z) = |{5, 11, 13}| = 3.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

For clarity, let us look at an example

We may take A = {5, 6, 10, 11, 12, 13, 18, 20, 22, 24, 28, 35} and P = {2, 7}. By sifting A with the given P, we see A2 = {6, 10, 12, 18, 20, 22, 24, 28}, and A7 = {28, 35} . We are left with S(A, P, z) = |{5, 11, 13}| = 3.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Other Brun Results

Twin Prime Conjecture Goldbach Conjecture

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Other Brun Results

Twin Prime Conjecture Goldbach Conjecture

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Twin Prime Conjecture

Conjecture The Twin Prime Conjecture states that there are infinitely many primes p such that p +2 is also prime. An example is (5,7). This is an unproven conjecture at this point; however, Brun used his sieve to show that the sum of the recipricals converges. Brun used his sieve to make progress on the conjecture by showing that there are infinitely many pairs of integers differing by 2, where each of the member of the pair is the product of at most 9 primes.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Twin Prime Conjecture

Conjecture The Twin Prime Conjecture states that there are infinitely many primes p such that p +2 is also prime. An example is (5,7). This is an unproven conjecture at this point; however, Brun used his sieve to show that the sum of the recipricals converges. Brun used his sieve to make progress on the conjecture by showing that there are infinitely many pairs of integers differing by 2, where each of the member of the pair is the product of at most 9 primes.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Twin Prime Conjecture

Conjecture The Twin Prime Conjecture states that there are infinitely many primes p such that p +2 is also prime. An example is (5,7). This is an unproven conjecture at this point; however, Brun used his sieve to show that the sum of the recipricals converges. Brun used his sieve to make progress on the conjecture by showing that there are infinitely many pairs of integers differing by 2, where each of the member of the pair is the product of at most 9 primes.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Goldbach Conjecture

This is one of the oldest unsolved problems in mathematics. Conjecture Every even integer greater than 2 is a Goldbach number, which is a number that can be expressed as two primes. For example: 2 + 2 = 4 3 + 3 = 6 3 + 5 = 8. Brun used his sieve to make progress on this conjecture as

  • well. He showed that very even number is the sum of two

numbers each of which is the product of at most 9 primes

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Goldbach Conjecture

This is one of the oldest unsolved problems in mathematics. Conjecture Every even integer greater than 2 is a Goldbach number, which is a number that can be expressed as two primes. For example: 2 + 2 = 4 3 + 3 = 6 3 + 5 = 8. Brun used his sieve to make progress on this conjecture as

  • well. He showed that very even number is the sum of two

numbers each of which is the product of at most 9 primes

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Goldbach Conjecture

This is one of the oldest unsolved problems in mathematics. Conjecture Every even integer greater than 2 is a Goldbach number, which is a number that can be expressed as two primes. For example: 2 + 2 = 4 3 + 3 = 6 3 + 5 = 8. Brun used his sieve to make progress on this conjecture as

  • well. He showed that very even number is the sum of two

numbers each of which is the product of at most 9 primes

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Goldbach Conjecture

This is one of the oldest unsolved problems in mathematics. Conjecture Every even integer greater than 2 is a Goldbach number, which is a number that can be expressed as two primes. For example: 2 + 2 = 4 3 + 3 = 6 3 + 5 = 8. Brun used his sieve to make progress on this conjecture as

  • well. He showed that very even number is the sum of two

numbers each of which is the product of at most 9 primes

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions

Goldbach Conjecture

This is one of the oldest unsolved problems in mathematics. Conjecture Every even integer greater than 2 is a Goldbach number, which is a number that can be expressed as two primes. For example: 2 + 2 = 4 3 + 3 = 6 3 + 5 = 8. Brun used his sieve to make progress on this conjecture as

  • well. He showed that very even number is the sum of two

numbers each of which is the product of at most 9 primes

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

The Famous Fibonacci Sequence

The Fibonacci Sequence is: Fn, defined by the recurrence relation: Fn = Fn−1 + Fn−2. They have seed values of F0 = 0 and F1 = 1. The first few terms are 1,1,2,3,5,8,13,21...

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

The Famous Fibonacci Sequence

The Fibonacci Sequence is: Fn, defined by the recurrence relation: Fn = Fn−1 + Fn−2. They have seed values of F0 = 0 and F1 = 1. The first few terms are 1,1,2,3,5,8,13,21...

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Brun Sieve and the Fibonacci Sequence

Let us take a finite amount of the Fibonacci sequence. A = Fn = {2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610} and let P = {2, 3, 5, 7, 11, ...}. After filtering using the set P, the primes, we are left with S(A, P, z) = |{2, 3, 5, 13, 89, 233}| = 6 . These are the prime Fibonacci numbers within this given Fn.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Fibonacci Primes

A Fibonacci number that is prime. Their finiteness is unknown. It has been calculated that the largest known Fibonacci prime is F81839, which has 17103 digits. It was proven to be such by Broadhurst and de Water in 2001.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Carmichael’s Theorem

Theorem Every Fibonacci number (aside from 1, 8, and 144) has at least

  • ne unique prime factor that has not been a factor of the

preceding Fibonacci numbers.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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1 1 2 3 5 8 13 21 34

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Fibonacci sequence

1 1 2 3 5 8 13 21 34 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 1 1 2 3 5 1 6 6

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Fibonacci sequence mod 7 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1...

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Theorem Let P be an arbitrary finite collection of primes. Then there exists a Fibonacci number that has no factors in P.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Finding Relative Primes Modulo 2: zeros every 3rd term Modulo 3: zeros every 4th term Modulo 7: zeros every 8th term 24th term: 46368 = 2 x 23184 = 3 x 15456 = 7 x 6624 25th term: 75025 ≡ 1 mod 2 ≡ 1 mod 3 ≡ 6 mod 7

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Finding Relative Primes Modulo 2: zeros every 3rd term Modulo 3: zeros every 4th term Modulo 7: zeros every 8th term 24th term: 46368 = 2 x 23184 = 3 x 15456 = 7 x 6624 25th term: 75025 ≡ 1 mod 2 ≡ 1 mod 3 ≡ 6 mod 7

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Finding Relative Primes Modulo 2: zeros every 3rd term Modulo 3: zeros every 4th term Modulo 7: zeros every 8th term 24th term: 46368 = 2 x 23184 = 3 x 15456 = 7 x 6624 25th term: 75025 ≡ 1 mod 2 ≡ 1 mod 3 ≡ 6 mod 7

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Finding Relative Primes Modulo 2: zeros every 3rd term Modulo 3: zeros every 4th term Modulo 7: zeros every 8th term 24th term: 46368 = 2 x 23184 = 3 x 15456 = 7 x 6624 25th term: 75025 ≡ 1 mod 2 ≡ 1 mod 3 ≡ 6 mod 7

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Finding Relative Primes Modulo 2: zeros every 3rd term Modulo 3: zeros every 4th term Modulo 7: zeros every 8th term 24th term: 46368 = 2 x 23184 = 3 x 15456 = 7 x 6624 25th term: 75025 ≡ 1 mod 2 ≡ 1 mod 3 ≡ 6 mod 7

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Distribution of Primes π(x) ∼ x log x Pprime ≈

x log x

x = 1 log x

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Distribution of Fibonacci Numbers lim

x→∞

Fn+1 Fn = φ Fn ≈ cφn n ≈ log x log const. PFibonacci ≈

log x const.

x

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Probability of Both Prime and Fibonacci Pprime·PFibonacci = 1 log x · log x x ·const. = 1 x ·const.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Sum over all natural numbers

  • x

P(P∩F) ≈

  • x

1 x → ∞

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Sum over all primes

  • p

P(P∩F) ≈

  • p

1 p → ∞

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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In conclusion, the Fibonacci primes appear to form an infinite set, but the argument is not valid since the sets are not independent.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

The idea of using a matrix was to allow us to easily see the prime Fibonacci numbers. We started off by taking an array with the Fibonacci numbers on top and the primes on the side. Then filled the array with the Fibonacci numbers modulo the primes.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

More Properties

                 2 3 5 8 13 21 34 55 89 · · · 2 1 1 1 1 1 1 3 2 2 2 1 1 1 2 5 2 3 3 3 1 4 4 7 2 3 5 1 6 6 6 5 11 2 3 5 8 2 10 1 1 13 2 3 5 8 8 8 3 11 17 2 3 5 8 13 4 4 4 19 2 3 5 8 13 2 15 17 13 23 2 3 5 8 13 21 11 9 20 . . . ...                 

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 82

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

We then reduced the array so that all the zeros on the diagonal represented the Prime Fibonacci numbers.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 83

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

                 2 3 5 8 13 21 34 55 89 · · · 2 1 1 1 1 1 1 3 2 2 2 1 1 1 2 5 2 3 3 3 1 4 4 11 2 3 5 8 2 10 1 1 13 2 3 5 8 8 8 3 11 29 2 3 5 8 13 21 5 26 2 37 2 3 5 8 13 21 34 18 15 59 2 3 5 8 13 21 34 55 30 89 2 3 5 8 13 21 34 55 . . . ...                 

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 84

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Then we created matrices from the array and looked at their properties A3 =   1 1 2 2 2 3   det(A3) = 10 rk(A3) = 3 A7 =             1 1 1 1 2 2 2 1 1 2 3 3 3 1 4 2 3 3 3 1 2 3 5 8 2 10 1 2 3 5 8 8 8 2 3 5 8 13 21 5 2 3 5 8 13 21 34             det(A7) = 0 rk(A7) = 6

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 85

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Then we created matrices from the array and looked at their properties A3 =   1 1 2 2 2 3   det(A3) = 10 rk(A3) = 3 A7 =             1 1 1 1 2 2 2 1 1 2 3 3 3 1 4 2 3 3 3 1 2 3 5 8 2 10 1 2 3 5 8 8 8 2 3 5 8 13 21 5 2 3 5 8 13 21 34             det(A7) = 0 rk(A7) = 6

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 86

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Then we created matrices from the array and looked at their properties A3 =   1 1 2 2 2 3   det(A3) = 10 rk(A3) = 3 A7 =             1 1 1 1 2 2 2 1 1 2 3 3 3 1 4 2 3 3 3 1 2 3 5 8 2 10 1 2 3 5 8 8 8 2 3 5 8 13 21 5 2 3 5 8 13 21 34             det(A7) = 0 rk(A7) = 6

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 87

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Then we created matrices from the array and looked at their properties A3 =   1 1 2 2 2 3   det(A3) = 10 rk(A3) = 3 A7 =             1 1 1 1 2 2 2 1 1 2 3 3 3 1 4 2 3 3 3 1 2 3 5 8 2 10 1 2 3 5 8 8 8 2 3 5 8 13 21 5 2 3 5 8 13 21 34             det(A7) = 0 rk(A7) = 6

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 88

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Then we created matrices from the array and looked at their properties A3 =   1 1 2 2 2 3   det(A3) = 10 rk(A3) = 3 A7 =             1 1 1 1 2 2 2 1 1 2 3 3 3 1 4 2 3 3 3 1 2 3 5 8 2 10 1 2 3 5 8 8 8 2 3 5 8 13 21 5 2 3 5 8 13 21 34             det(A7) = 0 rk(A7) = 6

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 89

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Then we created matrices from the array and looked at their properties A3 =   1 1 2 2 2 3   det(A3) = 10 rk(A3) = 3 A7 =             1 1 1 1 2 2 2 1 1 2 3 3 3 1 4 2 3 3 3 1 2 3 5 8 2 10 1 2 3 5 8 8 8 2 3 5 8 13 21 5 2 3 5 8 13 21 34             det(A7) = 0 rk(A7) = 6

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 90

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Theorem For all N ≥ 6, det(AN) = 0.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

As it turns out the last row is always a linear combination of the previous rows so, RNN − RNN−1 = (0, 0, ...0, a, b). a = 0 iff FN or FN−1 is a prime.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 92

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

If the Fibonacci primes are finite ∃N such that all the Fibonacci primes are ≤ FN. Hence ∀K > N, AK = AN ∗ ∗ B

  • Where TrB = K

i=N+1 Fi

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Conjecture For all N ≥ 6, rk(AN) = N − 1. A partial proof comes from the proof of the theorem. We have that rk(AN) ≤ N − 1 for all N ≥ 6.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 94

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Conjecture For all N ≥ 6, rk(AN) = N − 1. A partial proof comes from the proof of the theorem. We have that rk(AN) ≤ N − 1 for all N ≥ 6.

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory

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SLIDE 95

Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach

Thank you

Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory