Here is a computational check of the following special cases of the generalized Legendre conjecture:
The n5/3 conjecture.
For each positive integer n, there is a prime between n5/3 and (n+1)5/3.
The n8/5 conjecture. For each positive integer n, there is a prime between n8/5 and (n+1)8/5.
The n3/2 conjecture. For each integer n > 1051, there is a prime between n3/2 and (n+1)3/2.
The computation strongly suggests (but does not prove)
that the n5/3 and n8/5 conjectures hold for all positive n,
while the n3/2 conjecture fails for
n n5/3 < prime < (n+1)5/3 OK/fail n8/5 < prime < (n+1)8/5 OK/fail n3/2 < prime < (n+1)3/2 OK/fail