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A299103 Primes p = x^2 + y^2, not of form x^2 + 1, such that 2^(x^2) == 1 (mod p) or 2^(y^2) == 1 (mod p). 1
73, 433, 601, 673, 1801, 4513, 18433, 32377, 37633, 54001, 55201, 61681, 63901, 66529, 100801, 115201, 121369, 122921, 168781, 178481, 187417, 203617, 210913, 258721, 286721, 370661, 414721, 588061, 649657, 695701, 737537, 1781921, 3194101, 4674797, 4681801, 5039581, 6433561, 7593961, 7692697 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p = x^2 + y^2, not of form x^2 + 1, such that 2^(x^2) == 2 (mod p) or 2^(y^2) == 2 (mod p).

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..100

EXAMPLE

For 73 = 3^2 + 8^2 we have 2^9 == 1 (mod 73) and 2^64 == 2 (mod 73).

MAPLE

f:= proc(p) local F, x, y;

  if not isprime(p) then return false fi;

  if issqr(p-1) then return false fi;

  F:= GaussInt:-GIfactors(p)[2];

  x, y:= (Re, Im)(F[1][1]);

  2 &^ (x^2) mod p = 1 or 2 &^ (y^2) mod p = 1

end proc:

select(f, [seq(i, i=5..10^7, 4)]); # Robert Israel, Feb 02 2018

PROG

(PARI) B=bnfinit(x^2+1); { is_A299103(p) = my(z); if(p%4!=1 || issquare(p-1), return(0)); z=abs(Vec(bnfisintnorm(B, p)[1])); Mod(2, p)^(z[1]^2)==1 || Mod(2, p)^(z[2]^2)==1; } \\ Max Alekseyev, Feb 02 2018

CROSSREFS

Cf. A002313, A002496.

Sequence in context: A200908 A031418 A142229 * A201961 A142145 A227340

Adjacent sequences:  A299100 A299101 A299102 * A299104 A299105 A299106

KEYWORD

nonn

AUTHOR

Thomas Ordowski, Feb 02 2018

EXTENSIONS

Terms a(7) onward from Max Alekseyev, Feb 02 2018

STATUS

approved

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Last modified September 26 23:24 EDT 2020. Contains 337378 sequences. (Running on oeis4.)