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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 n^{5/3} < prime < (n+1)^{5/3} OK/fail n^{8/5} < prime < (n+1)^{8/5} OK/fail n^{3/2} < prime < (n+1)^{3/2} OK/fail

Copyright
© 2011, Alexei Kourbatov, JavaScripter.net.