A120336 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".

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, 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, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Giorgio Balzarotti and Paolo P. Lava, Jun 22 2006

Keywords

• nonn

Comments

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

Examples

			First primitive Pythagorean triad: 3, 4, 5.
Weierstrass equation: y^2 = x*(3^2 - x)*(4^2 + x).
Unique integer solution: (x,y) = (4,20).
First element in the sequence: 1.
Fifth primitive Pythagorean triad: 8, 15, 17.
Integer solutions: (x,y) = (15, 420) and (30, 510).
Fifth element in the sequence: 2.
		

Crossrefs

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

Programs

Maple

# 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;