A120336 - OEIS

%I #8 Nov 26 2015 00:19:26

%S 1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,

%T 1,1,2,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,2,1,1,1,1,1,

%U 1,1,1,1,1,1,2,1,1,1

%N Number of solutions (x,y) of Diophantine equation y^2 = x*(a^N - x)*(b^N + x) (Weierstrass elliptic equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg "a".

%C Triads a = 3, b = 4, c = 5 and a = 4, b = 3, c = 5 provide different results for (x,y).

%e First primitive Pythagorean triad: 3, 4, 5.

%e Weierstrass equation: y^2 = x*(3^2 - x)*(4^2 + x).

%e Unique integer solution: (x,y) = (4,20).

%e First element in the sequence: 1.

%e Fifth primitive Pythagorean triad: 8, 15, 17.

%e Integer solutions: (x,y) = (15, 420) and (30, 510).

%e Fifth element in the sequence: 2.

%p # a,b,c primitive Pythagorean triad n_sol:=0; for x from 1 by 1 to a^2 do y2:= x*( a^2 - x)*( x+ b^2); if ((floor(sqrt(y2)))^2=y2) n_sol:=n_sol+1;fi; print(n_sol) ; od;

%Y Cf. A009003, A020884, A120210, A120211, A120212, A120213.

%K nonn

%O 1,5

%A _Giorgio Balzarotti_ and _Paolo P. Lava_, Jun 22 2006