## Updated Elementary Proofs of Polignac Prime Conjecture, Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture

**Authors:** Stephen Marshall

This paper presents a complete and exhaustive proof of the Polignac Prime Conjecture. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer n:
n = (p-10!(1/p + ((-1)^d(d!))/(p+d)) + 1/p + 1/(p+d)
We use this proof for d = 2k to prove the infinitude of Polignac prime numbers.
The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Polignac Prime Conjecture possible.
Additionally, our proof of the Polignac Prime Conjecture leads to proofs of several other significant number theory conjectures such as the Goldbach Conjecture, Twin Prime Conjecture, Cousin Prime Conjecture, and Sexy Prime Conjecture. Our proof of Polignac’s Prime Conjecture provides significant accomplishments to Number Theory, yielding proofs to several conjectures in number theory that has gone unproven for hundreds of years.

**Comments:** 8 Pages. This is an update to my proff subitted in 2014, I have simpified the submission by removing uneccessary material from the proof.

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### Submission history

[v1] 2017-02-02 11:56:36

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