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Record (maximal) gaps between primes of the form 4k + 3.
3

%I #29 Jan 17 2019 02:55:18

%S 4,8,12,20,24,36,40,56,60,64,68,112,120,132,144,156,168,176,184,200,

%T 240,256,272,280,296,356,396,444,452,480,532,616,620,672,692,708,840,

%U 864,896,916,1004

%N Record (maximal) gaps between primes of the form 4k + 3.

%C Dirichlet's theorem on arithmetic progressions and GRH suggest that average gaps between primes of the form 4k + 3 below x are about phi(4)*log(x). This sequence shows that the record gap ending at p grows almost as fast as phi(4)*log^2(p). Here phi(n) is A000010, Euler's totient function; phi(4)=2.

%C Conjecture: a(n) < phi(4)*log^2(A268801(n)) almost always.

%C Conjecture: a(n) < phi(4)*n^2 for all n>2. - _Alexei Kourbatov_, Aug 12 2017

%H Alexei Kourbatov, <a href="http://arxiv.org/abs/1610.03340">On the distribution of maximal gaps between primes in residue classes</a>, arXiv:1610.03340 [math.NT], 2016.

%H Alexei Kourbatov, <a href="https://arxiv.org/abs/1709.05508">On the nth record gap between primes in an arithmetic progression</a>, arXiv:1709.05508 [math.NT], 2017; <a href="https://doi.org/10.12988/imf.2018.712103">Int. Math. Forum, 13 (2018), 65-78</a>.

%H Alexei Kourbatov and Marek Wolf, <a href="http://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

%e The first two primes of the form 4k+3 are 3 and 7, so a(1)=7-3=4. The next prime of this form is 11; the gap 11-7 is not a record so no term is added to the sequence. The next prime of this form is 19; the gap 19-11=8 is a new record, so a(2)=8.

%t re = 0; s = 3; Reap[For[p = 7, p < 10^8, p = NextPrime[p], If[Mod[p, 4] != 3, Continue[]]; g = p - s; If[g > re, re = g; Print[g]; Sow[g]]; s = p]][[2, 1]] (* _Jean-François Alcover_, Dec 12 2018, from PARI *)

%o (PARI) re=0; s=3; forprime(p=7, 1e8, if(p%4!=3, next); g=p-s; if(g>re, re=g; print1(g", ")); s=p)

%Y Cf. A002145, A084161.

%Y Corresponding primes: A268800 (lower ends), A268801 (upper ends).

%K nonn

%O 1,1

%A _Alexei Kourbatov_, Feb 13 2016