Re: update on recent post

[Max Alekseyev <maxale@gmail.com>]

Hi Randall,

What is called "area" in your message happens to be the area times 4.

As Charles pointed out, the actual areas are listed in https://oeis.org/A318575

Regards,

Max

On Wed, Sep 18, 2024 at 9:33 PM American Citizen <website.reader3@gmail.com> wrote:

To all:

John Ramsden and I have been working on trying to find the suitable
parameterization for a Heron triangle with all 3 sides square.

There are 2 known examples.

> area: 131674446875520 sides: a,b,c = 4427^2, 4380^2, 1853^2 gcd:1
> area: 1136958242123502720 sides a,b,c = 68595^2, 68104^2 ,11789^2 gcd:1
John determined that

> The actual equations defining a Heronian triangle with all sides
> square (when scaled by a suitable common factor) are the following
> simultaneous pair:
>
>   (1)   (a^2 + 1)/a = [ (b^2 + 1)/b ] c^2
>
>   (2)   (1 - a^2)/a + (1 - b^2)/b  = [ (b^2 + 1)/b ] d^2
where a,b,c,d are rationals (in Q)

We are hoping that we might be able to parameterize (1) in particular,
and use those results in (2) to help find such examples.

I wanted to update the GP Pari community as to our search status

Randall