## Definitive Proof of Legendre's Conjecture

**Authors:** Kenneth A. Watanabe

Legendre's conjecture, states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. In this paper, an equation was derived that accurately determines the number of prime numbers less than n for large values of n. Then, using this equation, it was proven by induction that there is at least one prime number between n^2 and (n + 1)^2 for all positive integers n thus proving Legendre’s conjecture for sufficiently large values n. The error between the derived equation and the actual number of prime numbers less than n was empirically proven to be very small (0.291% at n = 50,000), and it was proven that the size of the error declines as n increases, thus validating the proof.

**Comments:** 19 Pages.

**Download:** **PDF**

### Submission history

[v1] 2019-01-29 10:25:44

[v2] 2019-02-11 08:33:18

[v3] 2019-02-13 08:35:02

[v4] 2019-02-26 16:28:14

[v5] 2019-04-25 10:28:27

[v6] 2019-06-13 16:45:16

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