Don't you mean a,b,y are all > 1 ?
Yes, I did mean all > 1, sorry, didn't preview or check before posting
You seem to be looking to isolate each of x,a,b,y; here tizz:
sorry, no, that's not what I'm looking for because
this doesn't tell me if x is a whole number. Although it doen't sound like a big deal, to do this programatically, I would need to take the sqrt of (n^2 - aby), get the integer part of that, square it and compare to the original to see if it equaled (n^2 - aby), if not, go again. This takes a, relatively, long time. It would be much quicker if there was some way to 'know' that (n^2 - aby) was a square so that I would know that x was a whole number.
Here's one of the other formulas I have and what I done with it (not the same a and b btw)
a^2 + b^2 = 2c^2
[(a + b)/2]^2 + [(a - b)/2]^2 = c^2
let (a + b)/2 = u(u+2v)...(1)
and (a - b)/2 = 2v(u+v)...(2)
then I know that c is the whole number u^2 + 2uv + 2v^2 because (1)^2*(2)^2 = (u^2 + 2uv + 2v^2)^2
Adding (1) and (2) I get a = u^2 + 4uv + 2v^2
subtracting (1) - (2) I get b = u^2 - 2uv
so as long as u^2 > than 2uv, I know , for any whole numbers u and v, that a, b and c will also be whole numbers.
The answer that soroban gave is along the lines of what I'm looking for but there is a second formula that doesn't include a, b or c, that I can solve, but a third formula connects the other two.
Call the one above, the first one (it doesn't really matter what order they're taken in).
The second one is
d^2 - e^2 = 4f
and I done this
(d + e)(d - e) = 4f
let d + e = p
and d - e = q
then
d = (p+q)/2
e = (p-q)/2
f = pq/4
so the same as the first, for any whole p and q, I know that 2d, 2e and 4f will be whole numbers.
The third formula is
g^2 = e^2 + a^2 - c^2
so this has a and c in common with the first and e in common with the second. I was hoping to solve this and be able to p and q in terms of u and v to connect all three
putting in the values I've already "solved"
g^2 =((p-q)/2)^2 + 4uv(u+v)(u+2v)........................<------------- where my op came from
Soroban's answer was similar to what I'd already done for the second
(g + (p-q)/2)(g - (p-q)/2) = 4uv(u+v)(u+2v)
let g + (p-q)/2 = 4uv(u+v)
and g - (p-q)/2 = (u+2v)
then
g = (4uv(u+v) + (u+2v)) / 2
(p-q)/2 = (4uv(u+v) - (u+2v))/2
this gives me p - q = 4uv(u+v) - (u+2v)
so p = 4uv(u+v) - (u+2v) + q
for d, e and f I now have
d = (4uv(u+v) - (u+2v) + 2q)/2
e = (4uv(u+v) - (u+2v) )/2
f = (4uv(u+v) - (u+2v) + q)q/4
I can't find a way to get q in terms u and v to eliminate it from the values for d and f