A120210 Integer squares y from the smallest solutions of y^2 = x*(a^N - x)*(b^N + x) (elliptic line, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a.

20, 30, 156, 600, 420, 1640, 3660, 520, 2590, 7140, 1224, 10920, 8190, 20880, 32580, 4872, 19998, 5220, 48620, 69960, 3150, 41470, 97656, 132860, 19080, 76830, 176820, 230880, 131070, 12740, 296480, 11100, 375156, 52360, 209950, 468540, 64080

Offset: 1

Author

Giorgio Balzarotti, Paolo P. Lava, Jun 10 2006

Keywords

• nonn

Comments

The case x congruent to 0 mod b or b congruent to 0 mod x is frequent (e.g., A120212). Note that the triples a = 3, b = 4, c = 5 and a = 4, b = 3, c = 5 provide a different result for (x, y).

The natural solution is y = c * b * (c-b) and x = b * (c-b) with c hypotenuse in the triple. - Giorgio Balzarotti, Jul 19 2006

Examples

			First primitive Pythagorean triple: 3, 4, 5.
Weierstrass equation: y^2 = x*(3^2 - x)*(4^2 + x).
Smallest integer solution: (x, y) = (4,20).
First element in the sequence: y = 20.
		

References

• G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 47.

Crossrefs

Cf. A009003, A020884, A120211-A120213.

Programs

Maple

flag:=1; x:=0; # a, b, c primitive Pythagorean triple
while flag=1 do x:=x+1; y2:=x*(a^2-x)*(x+b^2); if (floor(sqrt(y2)))^2=y2 then print(sqrt(y2)); flag:=0; fi; od;