A120210 - OEIS

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

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OFFSET

1,1

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..37.

EXAMPLE

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.

MAPLE

flag:=1; x:=0; # a, b, c primitive Pythagorean triple