login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A299103 Primes p = x^2 + y^2, not of the form z^2 + 1, such that 2^(x^2) == 2^(y^2) == 1 (mod p). 1

%I #17 Oct 20 2020 13:54:19

%S 73,433,601,673,1801,4513,18433,32377,37633,54001,55201,61681,63901,

%T 66529,100801,115201,121369,122921,168781,178481,187417,203617,210913,

%U 258721,286721,370661,414721,588061,649657,695701,737537,1781921,3194101,4674797,4681801,5039581,6433561,7593961,7692697

%N Primes p = x^2 + y^2, not of the form z^2 + 1, such that 2^(x^2) == 2^(y^2) == 1 (mod p).

%H Max Alekseyev, <a href="/A299103/b299103.txt">Table of n, a(n) for n = 1..100</a>

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

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

%p if not isprime(p) then return false fi;

%p if issqr(p-1) then return false fi;

%p F:= GaussInt:-GIfactors(p)[2];

%p x,y:= (Re,Im)(F[1][1]);

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

%p end proc:

%p select(f, [seq(i,i=5..10^7,4)]); # _Robert Israel_, Feb 02 2018

%t pmax = 8000000;

%t xmax = Ceiling@ Sqrt@ pmax;

%t Table[p = x^2 + y^2; If[p <= pmax && PrimeQ[p] && (PowerMod[2, x^2, p] == 1 || PowerMod[2, y^2, p] == 1), p, Nothing], {x, 2, xmax}, {y, 2, x}] // Flatten // Union (* _Jean-François Alcover_, Oct 16 2020 *)

%o (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

%Y Cf. A002313, A002496.

%K nonn

%O 1,1

%A _Thomas Ordowski_, Feb 02 2018

%E Terms a(7) onward from _Max Alekseyev_, Feb 02 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)