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

0,1

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.

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.

Example

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].

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A104859 A108289 A007779 * A325294 A230960 A102038

Adjacent sequences: A084158 A084159 A084160 * A084162 A084163 A084164

Keyword

nonn

Author

Sven Simon, May 17 2003

Status

approved