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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