Positive integer solution to the elliptic curve y^2 = x^3 + 109 x^2 + 224 x

Ideas
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Sometime ago; I ran into the following problem discussed in:
(1.) How to find positive integer solutions to a/(b + c) + b/(a + c) + c/(a + b) = 4 . A more general problem namely; solving a/(b + c) + b/(a + c) + c/(a + b) = N is tackled in: (2.) An unusual cubic representation problem. The equality: a/(b + c) + b/(a + c) + c/(a + b) =
4 is called the: (3.) Fruit Cocktail problem . Using the following reference found in: (4.) Explicit Addition Formulae and considering: Y^2 + a1 X Y + a3 Y = X^3 + a2 X^2 + a4 X + a6 with:
a1 = 0, a2 = A, a3 = 0, a4 = B, and a6 = 0; you can use the analysis in (4.) to build a Mathematica module to handle point addition for elliptic curves of the form : y^2 = x^3 + A x^2 + B x. The attached Mathematica notebook gives an example of how to use this module to find a solution to (1.). A plot for the elliptic curve y^2 = x^3 + 109 x^2 + 224 x is also provided in the notebook.

Presentation.nb (90.1 KB)
Presentation.pdf (95.3 KB)

2

One solution:

FindInstance[y^2 == x^3 + 109 x^2 + 224 x && x > 0 && y > 0, {x, y}, Integers, 1, RandomSeeding -> Automatic]

(* {{x -> 4, y -> 52}} *)
1 Like
3

Yes, indeed! FindInstance[ x^3 + 109 x^2 + 224 x - y^2 == 0 && x > 0 && y > 0, {x, y}, Integers]
gives {x,y}= {4,5} as a solution. I didn’t know that RandomSeeding was available for FindInstance. Thank you sharing your solution Mariusz!