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Golden Oaks Software New Method to Find Primes GOLDEN OAKS SOFTWARE Description of a New Method to Find Primes Golden Oaks Software 7524 Soquel Way Citrus Heights, CA 95610 Dr. Joe Butler [email protected] (916) 220 1654 30 September 2008 Copyright Date: Oct 06, 2008 Patent Date: Oct 08, 2008 Number: 61195490 PROPRIETARY STATEMENT This document contains commercial or financial information, or trade secrets, of Golden Oaks Software, which are proprietary and exempt from disclosure to the public under the Freedom of Information Act, 5 U.S.C. 552(b)(4), and unlawful disclosure thereof is a violation of the Trade Secrets Act, 18 U.S.C. 1905. Public disclosure of any such information or trade secrets shall not be made without the written permission of Golden Oaks Software. This document includes data that shall not be disclosed outside the Government and shall not be duplicated, used or disclosed, in whole or in part, for any purposes other than to evaluate the information. If however, a contract is awarded to this offeror as a result of, or in conjunction with, the submission of this data, the Government shall have the right to duplicate, use, or disclose, the data to the extent provided in the resulting contract. The restriction does not limit the Government’s right to use information contained in this data if it is obtained from other sources without restriction. The data subject to this restriction is contained in all sheets. primefnd.doc 1 11/22/2008 COPYRIGHT & PATENTED New Method to Find Primes ABSTRACT The following are documents and word description flowcharts of a new method to find prime numbers. These algorithms allow a user to input a number and discover if that number is prime. The documents and flowcharts also show how to compute and verify the following: The twin number conjecture is true and infinite. The Strong Goldbach conjecture is true and infinite. The Weak Goldbach conjecture is true and infinite, but with new rules. The Weak Polignac conjecture is true and infinite. There are infinitely many primes of the form: n2 + 1. Fibonacci primes are infinite. There is a prime number between n2 and (n+1)2 for every positive integer n. The Brocard conjecture is true and infinite. primefnd.doc 2 11/22/2008 COPYRIGHT & PATENTED New Method to Find Primes DESCRIPTION OF NEW METHOD TO FIND PRIMES Technical Field The present invention relates generally to find primes in a simplified manner. Description of the Related Art How to find primes and prove primality. How does one go about finding primes? And once one has found them, how does one prove they are truly prime? Here, in this white paper, is presented the appropriate answers in several sections. Methods of determining prime numbers Many methods throughout the centuries, have been used for determining prime numbers. They include the following: Sieve of Eratosthenes Sieve of Atkin AKS primality test Fermat primality test Lucas-Lehmer test Solovay-Strassen primality test Miller-Rabin primality test Elliptic curve primality proving (ECPP) Diophantine equations Wilson's theorem Fermat's little theorem There are other methods that have been used to determine prime numbers, but the above are the main methods to date. primefnd.doc 3 11/22/2008 COPYRIGHT & PATENTED New Method to Find Primes Properties of primes At present, these are the established properties of prime numbers: When written in base 10, all prime numbers except 2 and 5 end in 1, 3, 7 or 9. (Numbers ending in 0, 2, 4, 6 or 8 represent multiples of 2 and numbers ending in 0 or 5 represent multiples of 5.) If p is a prime number and p divides a product ab of integers, then p divides a or p divides b. This proposition was proved by Euclid and is known as Euclid's lemma. It is used in some proofs of the uniqueness of prime factorizations. The ring Z/pZ is a field if and only if p is a prime. Put another way: p is prime if and only if φ(p) = p − 1. If p is prime and a is any integer, then ap − a is divisible by p (Fermat's little theorem). 1 If p is a prime number other than 2 and 5, /p is always a recurring decimal, whose period 1 is p − 1 or a divisor of p − 1. This can be deduced directly from Fermat's little theorem. /p expressed likewise in base q (other than base 10) has similar effect, provided that p is not a prime factor of q. An integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p (Wilson's theorem). Conversely, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. If n is a positive integer greater than 1, then there is always a prime number p with n < p < 2n (Bertrand's postulate). Adding the reciprocals of all primes together results in a divergent infinite series. More precisely, if S(x) denotes the sum of the reciprocals of all prime numbers p with p ≤ x, then S(x) = ln ln x + O(1) for x → ∞. In every arithmetic progression a, a + q, a + 2q, a + 3q, … where the positive integers a and q are coprime, there are infinitely many primes (Dirichlet's theorem on arithmetic progressions). The characteristic of every field is either zero or a prime number. If G is a finite group and pn is the highest power of the prime p which divides the order of G, then G has a subgroup of order pn. (Sylow theorems.) If G is a finite group and p is a prime number dividing the order of G, then G contains an element of order p. (Cauchy Theorem) The prime number theorem says that the proportion of primes less than x is asymptotic to 1 /ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x). The Copeland-Erdős constant 0.235711131719232931374143…, obtained by concatenating the prime numbers in base ten, is known to be an irrational number. primefnd.doc 4 11/22/2008 COPYRIGHT & PATENTED New Method to Find Primes The value of the Riemann zeta function at each point in the complex plane is given as a meromorphic continuation of a function, defined by a product over the set of all primes for Re(s) > 1 If p > 1, the polynomial is irreducible over Z/pZ if and only if p is prime. An integer n is prime if and only if the nth Chebyshev polynomial of the first kind Tn(x), divided by x is irreducible in Z[x]. Also if and only if n is prime. All prime numbers above 3 are of the form 6n − 1 or 6n + 1, because all other numbers are divisible by 2 or 3. Generalizing this, all prime numbers above q are of form q#·n + m, where 0 < m < q, and m has no prime factor ≤ q. primefnd.doc 5 11/22/2008 COPYRIGHT & PATENTED New Method to Find Primes Open Questions There are many open questions about prime numbers. A very significant one is the Riemann hypothesis, which essentially says that the primes are as regularly distributed as possible. From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about 1/ log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct, in particular, the simplest assumption is that primes should have no significant irregularities without good reason. Many famous conjectures appear to have a very high probability of being true (in a formal sense, many of them follow from simple heuristic probabilistic arguments): Prime Euclid numbers: It is not known whether or not there are an infinite number of prime Euclid numbers. Strong Goldbach conjecture: Every even integer greater than 2 can be written as a sum of two primes. Weak Goldbach conjecture: Every odd integer greater than 5 can be written as a sum of three primes. Twin prime conjecture: There are infinitely many twin primes, pairs of primes with difference 2. Polignac's conjecture: For every positive integer n, there are infinitely many pairs of consecutive primes which differ by 2n. When n = 1 this is the twin prime conjecture. A weaker form of Polignac's conjecture: Every even number is the difference of two primes. It is widely believed there are infinitely many Mersenne primes, but not Fermat primes. It is conjectured there are infinitely many primes of the form n2 + 1. Many well-known conjectures are special cases of the broad Schinzel's hypothesis H. It is conjectured that there are infinitely many Fibonacci primes. Legendre's conjecture: There is a prime number between n2 and (n + 1)2 for every positive integer n. Cramér's conjecture: . This conjecture implies Legendre's, but its status is more unsure. Brocard's conjecture: There are always at least four primes between the squares of consecutive primes greater than 2. All four of Landau's problems from 1912 are listed above and still unsolved: Goldbach, twin primes, Legendre, n2+1 primes. primefnd.doc 6 11/22/2008 COPYRIGHT & PATENTED New Method to Find Primes BRIEF SUMMARY OF THE INVENTION It is a primary object of the present invention to show that the order of prime numbers after 5, is nothing but a binary tree function.
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