Russian Journal of Mathematical Research. Series A, 2019, 5(1) 3 Copyright © 2019 by Academic Publishing House Researcher s.r.o. Published in the Slovak Republic Russian Journal of Mathematical Research. Series A Has been issued since 2015. E-ISSN: 2413-7529 2019, 5(1): 3-9 DOI: 10.13187/rjmr.a.2019.1.3 www.ejournal30.com Proofs of Andrica and Legendre Conjectures Samuel Bonaya Buya a , * a Ngao girls’ secondary school, Tana River County, Kenya Abstract In this research proof of Legendre conjecture is presented. The proof is based on a property possessed exclusively by all prime numbers. That is, the positive square-root of any prime number is an irrational number that always lies between two consecutive positive integers. This property excludes the number one from the set of prime numbers. Not all composite numbers possess this sure property possessed by all prime numbers. It is this special property of prime numbers special property of prime numbers that makes Legendre conjecture a sure law for all prime numbers. In the process of seeking to prove Legendre’s conjecture the prime gap problem is resolved and Riemann hypothesis reviewed in the light of these findings. Keywords: proof of Legendre’s conjecture, an exclusive property of prime numbers, number theory, Riemann hypothesis, on differences between consecutive primes. 1. Introduction The properties of prime numbers have been studied for many centuries. Euclid gave the first proof of infinity of primes. Euler gave a proof which connected primes to the zeta function. Then there was the Gauss and Legendre’s formulation of the prime number theorem and its proof by Hadamard and de la Vallee Poussin. Riemann further came with some hypothesis about the roots of the Riemann-zeta function. Many others have contributed towards prime number theory. Legendre’s conjecture, proposed by Adrien-Marie Legendre states that there is a prime number between n 2 and (n+1) 2 for every positive integer n. The conjecture is one of Landau’s problems (1912) on prime numbers. Up to 2017 the conjecture had neither been proved nor disproved. In this research a method will be presented of proving Legendre’s conjecture. The proof is based on an exclusive properties of prime numbers not generally shared with composite numbers. A square root property of prime numbers will be discussed that also implies the truthfulness of Legendre’s conjecture. Relevance The research aims at furthering our understanding of the prime gap problem as proposed in Legendre, Oppermann, Andrica conjectures and even Riemann hypothesis. * Corresponding author E-mail addresses: sbonayab@gmail.com (S. Bonaya Buya)