Problems & Puzzles: Conjectures

Conjecture 77. Gaps between primes of the form p=qn + r

Alexei Kourbatov sent the following conjecture for primes p=qn+r, gcd(q,r)=1.

Conjecture:
Gaps between primes p = qn + r up to x are less than phi(q)*log^2(x)
Here phi(q) is Euler's totient function, the number of positive integers <= q coprime to q. (This is a generalization of Cramer's conjecture. We get Cramer's case if q=2, r=1.)

No counterexamples exist for any 1<=r < q<=50, gcd(q,r)=1; x<10^10.

Example: q=10 r=1
The primes 180666691 and 180667801 both have the form 10n+1.
In between, there are no other primes of the form 10n+1.
The gap 1110 = 180667801 - 180666691 is less than phi(10)*log^2(180667801) = 4*(19.012)^2 = 1445.85
(NOTE: the logarithm is taken of the larger end of the gap.)

Other examples can be found in the OEIS:
A268925  Record (maximal) gaps between primes of the form 6k + 1.
A268928  Record (maximal) gaps between primes of the form 6k + 5.
A268799  Record (maximal) gaps between primes of the form 4k + 3.

A quick update: still no counterexamples. More data in the OEIS now:

A268984  Record (maximal) gaps between primes of the form 10k + 1.
A269234  Record (maximal) gaps between primes of the form 10k + 3.
A269238  Record (maximal) gaps between primes of the form 10k + 7.
A269261  Record (maximal) gaps between primes of the form 10k + 9.
A268925  Record (maximal) gaps between primes of the form 6k + 1.
A268928  Record (maximal) gaps between primes of the form 6k + 5.
A084162  Record (maximal) gaps between primes p = {1, 2} modulo 4.
A268799  Record (maximal) gaps between primes of the form 4k + 3.

See a convenient query to see the relevant OEIS sequences:

Weaker conjectures:

(I) Almost all maximal gaps between primes p = qn + r below x are less than phi(q)*log^2(x)
(II) Gaps between primes p = qn + r below x are O(phi(q)*log^2(x)).

These conjectures are based on the following "ingredients":
- the prime number theorem;
- Dirichlet's theorem on arithmetic progressions;
- a heuristic application of extreme value theory.

Q. Prove these conjectures or find counterexamples.

A. Kourbatov wrote on Set 19, 2017

Regarding Conjecture 77 - I have found an exceptional case:
For q=1605, r=341 (also q=3210, r=341), we have the primes
3415781 = 3624431 = 341 (mod 1605),  phi(1605)=phi(3210)=848,

and the gap G_{q,r} between these primes is 208650 > phi(q) log^2 (3624431) = 193434.64...

The "almost always" version of conjecture 77 seems very plausible; exceptions like the above are extremely rare (I do not know of any other exceptions at the moment).

This and related conjectures are included in arXiv:1610:03340

(The paper is not updated since January 2017, so it does not currently tell about this particular exception.)

...

By now I have found these counterexamples to Conjecture 77:

gap    prime1  prime2
208650 3415781 3624431 q=3210 r=341

316790 726611  1043401 q=4010 r=801

229350 1409633 1638983 q=4170 r=173

I do believe that the "almost always" version of conjecture 77 is true. (Last year I already updated the corresponding OEIS entries to state "almost always" in conjectures.)

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