As a postscript to a paper on the Tarry-Escott problem, Smyth [smy1] mentions that the elliptic curve he considers is a member of a family of curves linked to the following problem:
 

Find two rational right-angled triangles on the same base whose heights are in the ratio N:1.
 

He states that the late John Leech informed him of this connection. The following note discusses this problem. A much fuller discussion can be found in a technical report (Postscript version or PDF version ).

By scaling, we can consider the sides to be integers. Thus, let b be the common base, and the two heights h and Nh. Then, we have the quadratic integer equations

b2  +  h2  =  c2

b2  +  ( N  h )2  =  d 2

so that there is a direct link to the idea of concordant forms .

We can assume that b and h have no common factors, so either b = p² - q² ,
h = 2 p q or b = 2 p q ,  h = p² - q² , with p and q integers of opposite parity and no common factor.

Taking the first option and substituting into (2), we get

d2  =  p4  +  ( 4 N2 - 2 ) p2 q2  +  q4

which, defining e = d / q² , f = p / q , becomes

e2  =  f4  +  ( 4 N2 - 2 ) f2  +  1

This equation has an obvious rational solution, and so is birationally equivalent to an elliptic curve.

Using standard methods, we find the family of curves

y2  = x3  +  ( N2 + 1 ) x2  +  N2 x

with f  =  p / q  =  y / ( N² + x ).

If we assume that b = 2 p q , h = p² - q², we derive the same elliptic curve, but with the transformation p / q =  y / ( N x + N ).

As an example, for N = 28, we find that x = 112 / 9,  y = 9856 / 27 lies on the curve y² = x³ + 785 x ² + 784 x. This gives p / q = 11 / 24 and so b = 455 (ignoring - sign) and h = 528.

We can show that the points of finite order are

1. The point at infinity,
2. Points with x = 0 , -1 , -n² and y=0,
3. Points with x = N and y = ± N ( N + 1 ),
4. Points with x = -N and y = ±N ( N - 1 )

None of the points of finite order lead to a non-trivial solution of the original problem. Thus, we will only find a solution for those curves with rank > 0.

Birch and Swinnerton-Dyer conjecture calculations were done to estimate the rank for 1 £ N £ 999. For those curves with rank > 0, we then calculated a rational point of infinite order for practically all possible values of N. The resulting values of N, b, and h can be found in the solution file .

As of 25 April 2005, all the unsolved values of N in [2,999] have been resolved. Thanks to Randall Rathbun for the final N=502 value.

I might extend the values further, but not in the immediate future.