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Old 2020-08-08, 12:52   #27
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Dec 2008
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Originally Posted by mart_r View Post
It seems I was even more lucky than I originally thought with that gap at q=152. It gets increasingly hard to find gaps even with CSG>0.9.
Specifically, between 5*1012 and 7.3*1012 for q<=1000 the largest CSG is about 0.8849, which puts a large question mark behind ever finding another gap that is larger than both q² and \varphi(q) * log²p. I'm kind of heuristically challenged there.

From what I gather, the conjectures of Granville et al. for gaps between consecutive primes only, and at best, apply to prime gaps in AP with common difference q if the bounding prime p' = p+q*k is no larger than (p+n)² for very small n (n<2, say).
On the other hand, primes in AP may behave more random, as it were, than the primes themselves. Even if the gaps with CSG>1 is finite for any fixed q, is it plausible to expect that CSG can be arbitrarily large, maybe even as a function of q, how do you write it, f(q) > O(1)?

But I won't leave unanswered questions here today without providing a new record gap with both (q, r) prime:
p = 7,302,961,447
q = 214,451
r = 47,093
k = 576
g/\varphi(q)/log²p' = 1.1150385797428...
CSG (via sum Ri') = 1.1158450619299...
Which is also quite remarkable as it's in the upper decile of all gaps with CSG>1.
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