OFFSET
0,3
COMMENTS
For n>2, all other solutions (x,y) are divisible by 13, e.g., 26^2+39^2=13^3.
LINKS
Robert Israel, Table of n, a(n) for n = 0..3581
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Zak Seidov, Apr 10 2011
STATUS
approved