Quote:
Originally Posted by Crook
Let P(n) denote the nth prime number. Then, does anybody have an idea why P(n+1)<(sqrt(P(n))+1)^2 is true?

There is a famous
Legendre's conjecture AKA the 3rd Landau's problem that there is always a prime between n^2 and (n+1)^2 but I'm not sure if n is required to be integer. If n may be a positive real number then this conjecture directly implies your inequality. Otherwise, it implies a weaker inequality P(n+1)<(ceil(sqrt(P(n))+1)^2.