|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|