[freemind]

Determine a subset having different elements, so that the sum of the squares of its elements equals their product.
Do not simply post the subset, show how you found it.

[scorpius119]

an answer

how to arrive at it

If is a solution, so is . So if we can find any solution in positive integers, we should be able to use this to get one where they're all distinct.

By AM-GM, , so we'll need . The first place to look will be so that , or . Here we find a solution .

So is a solution. Bump up 3 to and then 1 to to obtain the above answer.

[campos]

i found and .

suppose we have a solution, and consider it as a quadratic in . we have its discriminant equals . it has to be a perfect square so .

suppose that and . so, we have that .

consider it again as a quadratic in . we have its discriminant equals . again we take
and , so, .

consider it again as a quadratic in . its discriminant equals . take . we have to find a integer such that
. for we have that .

if we return to previous equations, we find that , so we find ... analogously we find . if we find . if we find , and that's how i found them

[Euclid's Turtle]

campos, how do you get that discriminant equation? and why must it be a perfect square? To have 2 real roots, but in the same spot?

[campos]

the discriminant of the equation equals ...

it has to be a perfect square in order for to be an integer...