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Positive integers m such that m^4 = a^2 + b^2 and a + b = c^2 for some coprime integers a, b, c.
4

%I #16 Oct 18 2023 12:26:23

%S 1,13,1525,2165017,42422452969,7658246457672229,

%T 15512114571284835412957,452005526897888844293504165425,

%U 126314830357375266295717376544111167953,368440923990671763222767414151367493861848396861

%N Positive integers m such that m^4 = a^2 + b^2 and a + b = c^2 for some coprime integers a, b, c.

%C Values of m in coprime solutions to 2m^4 = c^4 + d^2 (so that a, b = (c^2 +- d)/2).

%C Corresponding values of c are given in A167437.

%C Terms with positive a,b,c are given in A166930.

%C This is a generalized Somos-4 sequence. - _Michael Somos_, Jan 29 2023

%H Keith Conrad, <a href="https://kconrad.math.uconn.edu/ross2008/descent2.pdf">Fermat's Big Problem</a>, 2008, slides 20-30.

%F a/m^2 = (-8*u^2 + 32*v + u^4 + 4*v^2 - 4*u^3 + 16*u*v)/(8 + u^2 + 4*u)^2 and b/m^2 = (4*u^3 - 8*u^2 - 4*v*u^2 - 16*u*v + 64)/(8 + u^2 + 4*u)^2 where (u,v) is a rational point on the elliptic curve v^2 = u^3 + 8*u.

%F a(n) = a(1-n) = (1764*a(n-1)*a(n-3) - 3107*a(n-2)^2)/a(n-4) = A360187(2*n-1) for all n in Z. - _Michael Somos_, Jan 29 2023

%o (PARI) {a(n) = my(A); if(n<1, n=1-n); A = vector(max(4, n+2)); A[1] = 13; A[2] = 1; A[3] = 1; A[4] = 13; for(k=5, n+2, A[k] = (1764*A[k-1]*A[k-3] - 3107*A[k-2]^2)/A[k-4]); A[n+2]}; /* _Michael Somos_, Jan 29 2023 */

%Y Cf. A360187.

%K nonn

%O 1,2

%A _Max Alekseyev_, Oct 23 2009

%E Edited by _Max Alekseyev_, Nov 03 2009