A268799 Record (maximal) gaps between primes of the form 4k + 3.

4, 8, 12, 20, 24, 36, 40, 56, 60, 64, 68, 112, 120, 132, 144, 156, 168, 176, 184, 200, 240, 256, 272, 280, 296, 356, 396, 444, 452, 480, 532, 616, 620, 672, 692, 708, 840, 864, 896, 916, 1004

Offset: 1

Author

Alexei Kourbatov, Feb 13 2016

Keywords

• nonn

Comments

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.

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

Conjecture: a(n) < phi(4)*n^2 for all n>2. - Alexei Kourbatov, Aug 12 2017

Examples

			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.
		

Links

• Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
• Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
• Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Crossrefs

Cf. A002145, A084161.

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

Programs

Mathematica

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 *)

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)