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Primes that are the sum of two squares and which set a record for the gap to the next prime of that form.
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%I #37 May 16 2019 08:51:17

%S 2,5,17,73,113,197,461,1493,1801,9533,15661,16741,33181,39581,50593,

%T 180797,183089,1561829,1637813,2243909,4468889,4874717,7856441,

%U 10087201,12021029,12213913,18226661,148363637,292182097,320262253,468213937

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

%C 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.

%C The length of the gap can be found in A084162.

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

%H Charles R Greathouse IV, <a href="/A084161/b084161.txt">Table of n, a(n) for n = 0..42</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 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].

%t 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 *)

%o (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

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

%K nonn

%O 0,1

%A _Sven Simon_, May 17 2003