

A084161


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


7



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
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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]] (* JeanFrançois Alcover, Feb 20 2019, from PARI *)


PROG

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


CROSSREFS

Cf. A002313, A084160, A084162 (gap sizes), A268963 (endofgap 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



