I've read Jim Fougeron's comment about Andrica's conjecture. I'm not a mathematician, so please be benevolent with me, but his proof attempt led me the following reasoning:
Write Andrica's Conjecture
sqrt( p[n+1] ) - sqrt( p[n] ) < 1
in the gap form
g[n] < 2sqrt( p[n] ) + 1
Choose y such that q^2 is the square number the nearest to and less than p[n]
q^2 + y := p[n]
g[n] < 2sqrt( q^2 + y ) + 1
Now, there must exist a number w such that
2sqrt( q^2 + y ) + 1 = 2q + 2w + 1
Let us ignore 2w for now, and write a stronger condition
g[n] < 2q + 1
We note that 2q + 1 is the difference between (q+1)^2 and q^2.
If this inequality is true, then starting counting at square q^2 and fetching odd integers until a prime is encountered it is very likely to find a prime before the next square (q+1)^2.
Using logarithmic integral fuction to find how many prime should we find between two square numbers
N(q) := Li((q+1)^2) - Li(q^2)
it can be seen that N(q) is monotonically increasing and at least two primes are expected between any two squares that asymptotically makes true the stronger condition and, a fortiori, Andrica's conjecture.
Does that resemble a sound proof?
_ Matteo Vitturi.
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