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

1, 13, 1525, 2165017, 42422452969, 7658246457672229, 15512114571284835412957, 452005526897888844293504165425, 126314830357375266295717376544111167953, 368440923990671763222767414151367493861848396861

Offset: 1

Author

Max Alekseyev, Oct 23 2009

Keywords

• nonn

Comments

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

Corresponding values of c are given in A167437.

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

This is a generalized Somos-4 sequence. - Michael Somos, Jan 29 2023

Links

• Keith Conrad, Fermat's Big Problem, 2008, slides 20-30.
• G. Jacob Martens, Rational right triangles and the Congruent Number Problem, arXiv:2112.09553 [math.GM], 2021, see section 11.1 The recursive algorithm.

Crossrefs

Cf. A360187.

Programs

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 */

Sage

# The Recursive Algorithm of G. Jacob Martens (2021)
A166929, max, l = [], 5, 0
def Fermat(x,l):
    p, q = numerator(x), denominator(x)
    a, b, c = p*q, -1/2*(p^2-q^2), 1/2*(p^2+q^2)
    A166929.append(sqrt(c))
    n, m = a-b, sqrt(a+b)*sqrt(c)
    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)
    l = l+1
    if lA166929.sort()
A166929 # Gerry Martens, Jan 14 2025

Formula

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.

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

Extensions

Edited by Max Alekseyev, Nov 03 2009