## Definitive Proof of the Twin-Prime Conjecture

**Authors:** Kenneth A. Watanabe

A twin prime is defined as a pair of prime numbers (p1,p2) such that p1 + 2 = p2. The Twin Prime Conjecture states that there are an infinite number of twin primes. A more general conjecture by de Polignac states that for every natural number k, there are infinitely many primes p such that p + 2k is also prime. The case where k = 1 is the Twin Prime Conjecture. In this document, a function is derived that corresponds to the number of twin primes less than n for large values of n. Then by proof by induction, it is shown that as n increases indefinitely, the function also increases indefinitely thus proving the Twin Prime Conjecture. Using this same methodology, the de Polignac Conjecture is also shown to be true.

**Comments:** 13 Pages. Paper was modified to include future directions and additional references were added

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### Submission history

[v1] 2018-12-11 16:27:17

[v2] 2018-12-27 09:47:28

[v3] 2019-01-09 08:19:00

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