SEVERAL PROOFS OF THE TWIN PRIMES AND GOLDBACH CONJECTURES

James Constant

math@coolissues.com

Proof of Goldbach's Conjecture, the Prime Number Theorem, and Euclid's Logic Provide Proofs of the Twin Primes Conjecture. Proof of the Twin Primes Conjecture Provides Proof of Goldbach's Conjecture

Theorem

There are infinitely many twin primes.

Proof of the Twin Primes Conjecture Using Proofs of Goldbach's Conjecture or Using the Prime Number Theorem

The twin primes conjecture (TPC) suggests that there is an infinite number of primes a and b with a difference 2, i.e., a - b = 2. Goldbach's conjecture (GC) suggests that every even number greater than 2 is the sum s of two prime numbers a and b, i.e., a + b = s where s is even >2. GC is proved by the author herein below and elsewhere¹

For prime numbers a,b,c

. . . . . a - b = (a + c) - (b + c) . . . . . even integer . . . . . . . . . . (1)

and thus, generally,

. . . . . a - b = 2k . . . . . . . . . . . . . k = integer . . . . . . . . .. . . (2)

and since a + b is an even number

. . . . . a + b = 2n . . . . . . . . . . . . n = integer>2 . . . . . . . . . . . . . (3)

Now, using (2) and (3) results in

. . . . . a = n + k . . . and . . . b = n - k . . . n>k . . . . . . . . . . . . . (4)

which say that for every single value of k primes a and b are separated by an interval 2k and occur as numbers n + k and n - k. Suppose that n₁,n₂,n₃, . . . ,n_r are all the twin prime (k=1) intervals. Since number n is unlimited, intervals n₁,n₂,n₃, . . . ,n_r are not all the twin prime intervals, i.e., in n₁,n₂,n₃, . . . ,n_r

Because all numbers n + k and n - k are not prime numbers, equations (2)-(4) are necessary for proving the infinitude but are not sufficient for finding the existence of prime numbers at these locations. However, from the Prime Number Theorem (PNT)², I compute that the probability of finding twin primes a at n + 1 and b at n -1 is about 2/logn (k=1) and the number of twin primes in the interval n is about 2n/(logn)². Also, in contrast to the foregoing actual proof using Goldbach's conjecture, the PNT provides a proof of the TPC as .³ We can also say that

• the difference of two prime numbers a and b is an even number 2k; and
• the probability of finding primes a at n + k and b at n - k decreases as n increases.

Proof of the Twin Primes Conjecture Using Euclid's Logic

Another actual proof of the TPC is suggested by Euclid's proof that there are infinitely many primes.⁴

Suppose that i₁=i₂=i₃= . . . =i_r=2 are all the twin prime intervals. Let I_r =2^r +1, i_r+1 =2 and I_r+1 =2^r+1 +1 from which it follows that 2I_r= I_r+1+1. For all values of r, since interval i_r+1 divides 2I_r (without dividing I_r and I_r+1,) it is yet another interval and intervals i₁=i₂=i₃= . . . =i_r are not all the twin prime intervals, i.e., in i₁=i₂=i₃= . . . =i_r=2.

Proof of Goldbach's Conjecture Using Proof of the Twin Primes Conjecture

I can now prove Goldbach's Conjecture by reversing my foregoing Proof of the Twin Primes Conjecture Using Proof of Goldbach's Conjecture. With reference to equations (2)-(4), suppose that not every sum of two primes a and b is an even number 2n>4. However, since the number of twin primes is unlimited, their sum is an even number 2n unlimited and every sum of primes is an even number.

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¹ Algebraic Factoring of the Cryptography Modulus and Proof of Goldbach's Conjecture at http://goldbach.coolissues.com/goldbach.htm

² See Chris Caldwell's "Prime Number Theorem at http://primes.utm.edu/glossary/page.php?sort=PrimeNumberThm

³ The PNT is strongest for providing proofs of the infinite number of primes and is weakest for accurately finding the existence of primes at given locations n. For this reason it is labeled as an heuristic, less than certain, proof of the infinitude of primes.

⁴ See Chris Caldwell's "Twin Primes" at http://primes.utm.edu/top20/page.php?id=1

Copyright 2004 by James Constant

By the same author: http://www.coolissues.com/mathematics/sameauthor.htm