A230962 Values of x such that x^2 + y^2 = 73^n with x and y coprime and 0 < x < y.
3, 48, 296, 721, 10072, 213785, 1958709, 7613760, 21165597, 894454032, 12278087704, 59926173839, 62518379032, 3374316625735, 58552907681096, 416603004343680, 1261259807092797, 10231862403603888, 255781764375436389, 2697529798981443601, 11543491568219853608
Offset: 1
Keywords
Examples
a(3) = 296 because 296^2 + 549^2 = 389017 = 73^3.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- Chris Busenhart, Lorenz Halbeisen, Norbert Hungerbühler, and Oliver Riesen, On primitive solutions of the Diophantine equation x^2+ y^2= M, Eidgenössische Technische Hochschule (ETH Zürich, Switzerland, 2020).
Programs
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Maple
f:=n -> min([abs@Re,abs@Im]((3+8*I)^n)): map(f, [$1..50]); # Robert Israel, Mar 31 2017
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Mathematica
Table[Select[PowersRepresentations[73^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *) Table[Min[Abs[Re[(3 + 8I)^n]], Abs[Im[(3 + 8I)^n]]], {n, 30}] (* Indranil Ghosh, Mar 31 2017, after formula by Robert Israel *) -
Python
from sympy import I, re, im print([min(abs(re((3 + 8*I)**n)), abs(im((3 + 8*I)**n))) for n in range(1, 31)]) # Indranil Ghosh, Mar 31 2017, after formula by Robert Israel
Formula
From Robert Israel, Mar 31 2017: (Start)
a(n) = min(abs(Re((3+8i)^n)), abs(Im((3+8i)^n))).
a(n) = abs(Re(3+8i)^n) if and only if 1/4 < frac(n*arctan(8/3)/Pi) < 3/4.
(End)
Comments