On Sunday 24 April 2005 22:54, you wrote:

> As far as the twin primes we know, is it true that there is a pair between

> every set of integers between P and P squared, where P is any prime number

> > 2?

No, try 19.

? forprime(p=19^2,20^2,print(p))

367

373

379

383

389

397

There's also 53.

? forprime(p=53^2,54^2,print(p))

2819

2833

2837

2843

2851

2857

2861

2879

2887

2897

2903

2909

Nothing else up to 50000, and no other counterexamples are expected. That's

because heuristically, twin prime pairs in the vicinity of n occur about

every O(log^2 n) integers, and your interval has (n+1)^2 - n^2 = 2n + 1

integers. Since 2n + 1 grows faster than log^2 n, one would expect no

counterexamples to your statement other than these `small' ones. By the way,

numerical evidence using some better approximations (including constants and

so on) lends credence to this heuristic.

Décio

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