OFFSET
1,2
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.
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
PROG
(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 l<max:
if x1!=1 :
Fermat(x1, l)
if x2!=1 :
Fermat(x2, l)
return
Fermat(1, l)
A166929.sort()
A166929 # Gerry Martens, Jan 14 2025
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Max Alekseyev, Oct 23 2009
EXTENSIONS
Edited by Max Alekseyev, Nov 03 2009
STATUS
approved