A084161 Primes that are the sum of two squares and which set a record for the gap to the next prime of that form.

2, 5, 17, 73, 113, 197, 461, 1493, 1801, 9533, 15661, 16741, 33181, 39581, 50593, 180797, 183089, 1561829, 1637813, 2243909, 4468889, 4874717, 7856441, 10087201, 12021029, 12213913, 18226661, 148363637, 292182097, 320262253, 468213937

Offset: 0

Author

Sven Simon, May 17 2003

Keywords

• nonn

Comments

Real primes 2, 5, 13, 17, 29, 37, ... (A002313) have a unique representation as sum of two squares. Values larger than 2 are the primes p with p = 1 mod 4. If p = x^2 + y^2, the corresponding complex prime is x + y * i, where i is the imaginary unit.

The length of the gap can be found in A084162.

Examples

			a(3) = 73: There are no primes p = 1 mod 4 between 73 and 89, this gap is the largest up to 89, the length is 16. Note that 73 = (8 - 3i)(8 + 3i) and 89 = (8 - 5i)(8 + 5i). The primes 79 and 83 are inert in Z[i].
		

References

• Ervand Kogbetliantz and Alice Krikorian, Handbook of First Complex Prime Numbers, Parts 1 and 2, Gordon and Breach, 1971.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..42
• Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Crossrefs

Cf. A002313, A084160, A084162 (gap sizes), A268963 (end-of-gap primes).

Programs

Mathematica

Reap[Print[2]; Sow[2]; r = 0; p = 5; For[q = 7, q < 10^7, q = NextPrime[q], If[Mod[q, 4] == 3, Continue[]]; g = q - p; If[g > r, r = g; Print[p] Sow[p]]; p = q]][[2, 1]] (* Jean-François Alcover, Feb 20 2019, from PARI *)

PARI

print1(2);r=0;p=5;forprime(q=7,1e7,if(q%4==3,next);g=q-p;if(g>r,r=g;print1(", "p));p=q) \\ Charles R Greathouse IV, Apr 29 2014