Solutions to Ax^2 +By^2=(A+B)^(2n+1)

ramsey2879

No not the trival solution but generally GCD(x,y) = 1 when GCD(A,B)=1

[tex] x = \sum_{k=0}^{n}(-A)^k*B^{n-k}*\binom{2n+1}{2k+1}[/tex]

[tex] y = \sum_{k=0}^{n}(-B)^k*A^{n-k}*\binom{2n+1}{2k+1}[/tex]

My question is was this known before?

Example A=8, B = 3, n = 1

x = 1*3*3 - 8*1*1 = 1; y = 1*8*3 -3*1*1 = 21; 8*1^2 + 3*21^2 = 11^3

for n = 2, x = 131; y = 89; 8*131^2 + 3*89^2 = 11^5