Legendre's Conjecture: Exploring the Search for Primes Between n^2 and (n+1)^2

[CRGreathouse]

Is anything more known about Legendre's conjecture that there is a prime between n^2 and (n+1)^2 for positive integers n than what appears on MathWorld?

MW says that a prime or semiprime always satisfies this, and that there is always a prime between n and n^{23/42} (21/42 would be equivilent to Legendre's conjecture).

How far has this been checked? It seems 'obvious' that it should hold, and yet there's no clear method of attacking the problem.

 

[CRGreathouse]

Also, I see here
http://www.primepuzzles.net/problems/prob_004.htm

a mention of a conjecture of Schinzel: $$\pi(n+\ln(n)^2)>\pi(n)$$ for n > 8. Does anyone have a reference for this, or evidence of its correctness? It seems a lot sharper than many conjectures I've seen, and certainly it's sharper than Legendre's conjecture. Still, the number of primes in the regon seems to grow at a fair pace, leading me to believe that it's at least reasonable.

 

[camilus]

Is anything more known about Legendre's conjecture that there is a prime between n^2 and (n+1)^2 for positive integers n than what appears on MathWorld?

MW says that a prime or semiprime always satisfies this, and that there is always a prime between n and n^{23/42} (21/42 would be equivilent to Legendre's conjecture).

How far has this been checked? It seems 'obvious' that it should hold, and yet there's no clear method of attacking the problem.

Since we were on this subject, do you know where to find more information on this?

 

[CRGreathouse]

I don't know of any recent progress. Here's an article with some background:
http://projecteuclid.org/euclid.mmj/1029003189