Generalized Legendre Conjecture:
a partial computational verification

Here is a computational check of the following special cases of the generalized Legendre conjecture:

The n^5/3 conjecture. For each positive integer n, there is a prime between n^5/3 and (n+1)^5/3.
The n^8/5 conjecture. For each positive integer n, there is a prime between n^8/5 and (n+1)^8/5.
The n^3/2 conjecture. For each integer n > 1051, there is a prime between n^3/2 and (n+1)^3/2.

The computation strongly suggests (but does not prove) that the n^5/3 and n^8/5 conjectures hold for all positive n, while the n^3/2 conjecture fails for n = 10, 20, 24, 27, 32, 65, 121, 139, 141, 187, 306, 321, 348, 1006, and 1051. Additional checks for the first ten million values of n do not yield any other counterexamples. We observe that, as n grows larger, prime gaps become relatively smaller and smaller, as compared to the intervals [n^s, (n+1)^s] – in other words, although prime gaps do grow, the width of intervals [n^s, (n+1)^s] grows even faster. This makes additional counterexamples extremely unlikely for very large n.

      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 

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