Number Theory

    

Definitive Proof of the Twin-Prime Conjecture

Authors: Kenneth A. Watanabe

A twin prime is defined as a pair of prime numbers (px, py) such that px+2 =py. The Twin Prime Conjecture states that there are an infinite number oftwin primes. A more general conjecture by de Polignac states that for everynatural number k, there are infinitely many primes p such that p + 2k isalso prime. The case where k = 1 is the Twin Prime Conjecture. In thisdocument, the function Π2∗ (n) is derived that closely approximates Π2(n),the actual number of twin primes less than n, for large values of n. Thenby proof by induction on Π2∗(n) , it is shown that for any prime number pi,there is at least one twin prime (px, py) such that pi2< py < (pi+1)2. Sincethere are an infinite number of prime numbers pi, this proves that there arean infinite number of twin primes, thus proving the Twin Prime Conjecture.Error analysis shows that the maximum error between Π2∗(n) and Π2(n) increases at a slower rate than Π2∗(n). Using this same methodology, the de PolignacConjecture is also shown to be true.

Comments: 23 Pages. Error analysis was added and errors were corrected from previous version.

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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
[v4] 2019-01-23 08:21:08
[v5] 2025-08-06 21:19:52

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