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

1,1

Comments

The corresponding y-values are in A230963.

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).

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)

Example

a(3) = 296 because 296^2 + 549^2 = 389017 = 73^3.

Maple

f:=n ->  min([abs@Re, abs@Im]((3+8*I)^n)):
map(f, [$1..50]); # Robert Israel, Mar 31 2017

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 *)

Prog

(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

Crossrefs

Cf. A230963, A188948, A230622, A230644, A230710, A230712, A230743, A230759, A230841.

Sequence in context: A162333 A003878 A219056 * A073982 A356644 A354132

Adjacent sequences: A230959 A230960 A230961 * A230963 A230964 A230965

Keyword

nonn

Author

Colin Barker, Nov 02 2013

Status

approved