On Tuesday 09 November 2004 11:43, you wrote:
> Hello, Group.
> Is it safe to say that two twin-prime pairs will occur between n squared
> and (n+2 squared plus the natural log of n+2)... or n^2 < p1,p2,q1,q2 <
> [(n+2)^2 +ln(n+2)]?
I believe it's likely. See my post with an heuristic for the number of twin
prime pairs between n^2 and (n+2)^2: this heuristic indicates a growing
number of twin prime pairs as n grows. Enlarging the interval by log(n+2)
would only improve that. However, it can't be proved easily.
> I can see that 71, 73 and 101, 103 both appear between 64 and 104 as a
> narrowing ex-ample. Can someone find a counter-example?
Actually 10^2 + log(10) = 102.30, so I would say this is a counter-example.
Other than that, there's [18^2, 20^2 + log(20)] = [324,402.996], for which
only 347 and 349 exist. For [26^2, 28^2 + log(28)] = [676,787.33], no pairs
exist. For [30^2, 32^2 + log(32)] = [900,1027.47], only 1019 and 1021 exist.
For [121^2, 123^2 + log(123)] = [14641,15133.81], only 14867 and 14869 exist.
Above that, and for all primes up to 2e9 (the best I can do with PARI/GP),
which equates to n = 44719, I couldn't find any counterexamples. The number
of twin-prime pairs in the given interval is increasing nicely -- for n =
10000, there's ~150 of them. That's not far from what my heuristic predicts
(118). So I believe these are the last counterexamples you'll find.
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