Legendre's conjecture states that there is a prime \(p\) between \(n^2\) and \((n+1)^2\) for every positive integer \(n\). This conjecture has a very peculiar correlation with the prime number theorem. It seems that there is around \({n \over ln(n)}\) amount of primes between \(n^2\) and \((n+1)^2\). here's a small table I made showing what Im talking about. The n/ln(n) is VERY close to the number of primes between \(n^2\) and \((n+1)^2\). Please Register or Log in to view the hidden image! Number of primes between n^2 and (n+1)^2. 0, 2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 4, 6, 7, 5, 6, 6, 7, 7, 7, 6, 9, 8, 7, 8, 9, 8, 8, 10, 9, 10, 9, 10, 9, 9, 12, 11, 12, 11, 9, 12, 11, 13, 10, 13, 15, 10, 11, 15, 16, 12, 13, 11, 12, 17, 13, 16, 16, 13, 17, 15, 14, 16, 15, 15, 17, 13, 21, 15, 15, 17, 17, 18, 22, 14, 18, 23, 13 source: http://www.research.att.com/~njas/sequences/A000040 And with Tsutomu Hashimotos paper On a certain relation between Legendre's conjecture and Bertrand's postulate, he proved this wonderful relation concerning Legendre's conjecture and Bertrand's postulate. Bertrand's postulate is similar to Legendre's conjecture, it states that there is a prime number \(p\) between \(n\) and \(2n\) for every positive integer \(n\). His results are incredible (if you like the theory of numbers of course), he found that the prime-counting function \(\pi (n)\) applied to Legendre's conjecture and Bertrand's postulate gave the wonderful relationship, it follows since \(\pi \left ((n+1)^2 \right ) = \pi \left ( n^2 +2n \right )\), that \(\pi \left ((n+1)^2 \right ) - \pi (n^2) = \pi (2n) - \pi (n) + 1 - \Phi_T \left ( n^2, 2n, \pi (n) \right ) \) I'll stop here because I just had a brilliant idea based on Ramanujan's proof of Bertrand's postulate. Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image!
