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A230710 Values of x such that x^2 + y^2 = 5^n with x and y coprime and 0 < x < y.

%I A230710 #47 May 23 2021 10:19:42
%S A230710 1,3,2,7,38,44,29,336,718,237,2642,10296,8839,16124,108691,164833,
%T A230710 24478,922077,2521451,1476984,6699319,34182196,35553398,32125393,
%U A230710 306268562,597551756,130656229,2465133864,8701963882,6890111163,15949374758,98248054847,135250416961
%N A230710 Values of x such that x^2 + y^2 = 5^n with x and y coprime and 0 < x < y.
%C A230710 The corresponding y-values are in A230711.
%C A230710 For all non-coprime solutions (x,y) to the equation x^2 + y^2 = p^n, x and y are both divisible by the prime p.
%C A230710 Using de Moivre's Theorem (in essence), define (c,d)*(e,f) as (ce-df,cf+de). Then a(n) = min{|u(n)|, |v(n)|}, where (u(n),v(n)) = (2,1)^n = (2,1)*(2,1)^[n-1]. Proof: It can be readily seen that u^2(n) + v^2(n) = 5^n. To show that u(n) and v(n) are relatively prime, assume that x,y are relatively prime. Then (2,1)*(x,y) = (2x-y, x+2y). If a prime p were to divide both of 2x-y and x+2y, then p would divide 5y, so p=5. Now suppose x == 2 (mod 5) and y == 1 (mod 5). It can be seen that 2x-y == -2 (mod 5) and x+2y == -1 (mod 5). The reverse also holds. Because u(1)=2 and v(1)=1, the result follows inductively. - _Richard Peterson_, May 21 2021
%H A230710 Zak Seidov, <a href="/A230710/b230710.txt">Table of n, a(n) for n = 1..200</a>
%H A230710 Chris Busenhart, Lorenz Halbeisen, Norbert Hungerbühler, and Oliver Riesen, <a href="https://people.math.ethz.ch/~halorenz/publications/pdf/Miniatur.pdf">On primitive solutions of the Diophantine equation x^2+ y^2= M</a>, Eidgenössische Technische Hochschule (ETH Zürich, Switzerland, 2020).
%e A230710 a(4)=7 because 7^2 + 24^2 = 625 = 5^4.
%t A230710 Table[Select[PowersRepresentations[5^n, 2, 2], CoprimeQ[#[[1]], #[[2]]] &][[1,1]], {n, 33}] (* _T. D. Noe_, Nov 04 2013 *)
%Y A230710 Cf. A000351, A230711.
%Y A230710 Cf. A188948, A230622, A230644.
%Y A230710 Cf. A139011.
%K A230710 nonn
%O A230710 1,2
%A A230710 _Colin Barker_, Oct 28 2013
%E A230710 Typo in data fixed by _Colin Barker_, Nov 02 2013