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Old 2005-10-26, 21:11   #3
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Feb 2005

25210 Posts

Originally Posted by Crook
Let P(n) denote the n-th 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.
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