A158936 List of coprime pairs (x,y) such that x^2+y^2 = 13^n with 0

0, 1, 2, 3, 5, 12, 9, 46, 119, 120, 122, 597, 828, 2035, 4449, 6554, 239, 28560, 56403, 86158, 145668, 341525, 246046, 1315911, 3369960, 3455641, 3627003, 17021162, 23161315, 58317492, 128629846, 186118929, 13651680, 815616479, 1590277918, 2474152797, 4241902555, 9719139348, 6712571031, 37641223154, 95420159401, 99498527400, 107655263398, 485257533003

Offset: 0

Author

Zak Seidov, Apr 10 2011

Keywords

• nonn
• tabf

Comments

For n>2, all other solutions (x,y) are divisible by 13, e.g., 26^2+39^2=13^3.

Examples

			n=1: 13^1=2^2+3^2, hence a(1)=2, a(2)=3,
n=2: 13^2=5^2+12^2, hence a(3)=5, a(4)=12.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..3581

Crossrefs

Cf. A098122 for case x^2+y^2=5^n.

Cf. A188948, A188949 for the values of x and y separately.

Cf. A188982, A188983 for even and odd terms.

Programs

Maple

f:= proc(n) local q;
  q:= map(abs, [Re,Im]((2+3*I)^n));
  op(sort(q))
end proc:
map(f, [$0..50]); # Robert Israel, Feb 27 2024

Mathematica

s={2,3};x=2;y=3;Do[A=3x+2y;If[Mod[A,13]==0,A=Abs[3x-2y];B=2x+3y,B=Abs[2x-3y]];x=A;If[A>B,x=B;y=A,y=B];s=Join[s,{x,y}],{20}];s
Table[Select[PowersRepresentations[13^n, 2, 2], CoprimeQ @@ # &][[1]], {n, 0, 21}] (* T. D. Noe, Apr 12 2011 *)

Formula

a(2*n) = min(|Re((3+2*i)^n)|, |Im((3+2*i)^n)|), a(2*n+1) = max(|Re((3+2*i)^n)|, |Im((3+2*i)^n)|). - Robert Israel, Feb 27 2024