cp's OEIS Frontend

A166929 Positive integers m such that m^4 = a^2 + b^2 and a + b = c^2 for some coprime integers a, b, c.

%I A166929 #26 Jan 29 2025 11:55:19
%S A166929 1,13,1525,2165017,42422452969,7658246457672229,
%T A166929 15512114571284835412957,452005526897888844293504165425,
%U A166929 126314830357375266295717376544111167953,368440923990671763222767414151367493861848396861
%N A166929 Positive integers m such that m^4 = a^2 + b^2 and a + b = c^2 for some coprime integers a, b, c.
%C A166929 Values of m in coprime solutions to 2m^4 = c^4 + d^2 (so that a, b = (c^2 +- d)/2).
%C A166929 Corresponding values of c are given in A167437.
%C A166929 Terms with positive a,b,c are given in A166930.
%C A166929 This is a generalized Somos-4 sequence. - _Michael Somos_, Jan 29 2023
%H A166929 Keith Conrad, <a href="https://kconrad.math.uconn.edu/ross2008/descent2.pdf">Fermat's Big Problem</a>, 2008, slides 20-30.
%H A166929 G. Jacob Martens, <a href="https://arxiv.org/abs/2112.09553">Rational right triangles and the Congruent Number Problem</a>, arXiv:2112.09553 [math.GM], 2021, see section 11.1 The recursive algorithm.
%F A166929 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 A166929 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 A166929 (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 */
%o A166929 (Sage)
%o A166929 # The Recursive Algorithm of G. Jacob Martens (2021)
%o A166929 A166929, max, l = [], 5, 0
%o A166929 def Fermat(x,l):
%o A166929     p, q = numerator(x), denominator(x)
%o A166929     a, b, c = p*q, -1/2*(p^2-q^2), 1/2*(p^2+q^2)
%o A166929     A166929.append(sqrt(c))
%o A166929     n, m = a-b, sqrt(a+b)*sqrt(c)
%o A166929     x1, x2 = ((2*m*n)^2+n^4+4*m*n*sqrt(8*m^4+n^4))/(16*m^4+n^4),((2*m*n)^2+n^4-4*m*n*sqrt(8*m^4+n^4))/(16*m^4+n^4)
%o A166929     l = l+1
%o A166929     if l<max:
%o A166929         if x1!=1 :
%o A166929             Fermat(x1,l)
%o A166929         if x2!=1 :
%o A166929             Fermat(x2,l)
%o A166929     return
%o A166929 Fermat(1,l)
%o A166929 A166929.sort()
%o A166929 A166929 # _Gerry Martens_, Jan 14 2025
%Y A166929 Cf. A360187.
%K A166929 nonn
%O A166929 1,2
%A A166929 _Max Alekseyev_, Oct 23 2009
%E A166929 Edited by _Max Alekseyev_, Nov 03 2009