# list of “Pythagorean triple” equations for $a^2+b^2=2c^2$

I have come up with an equation recently from a project that I'm doing. it's like the Pythagorean theorem except it's $$2c^2$$ rather than the familiar $$c^2$$, then I decided that I wanted a list of all the equations that could give me all "Pythagorean Triples", but I don't know where to start (except guess and check).

If anyone is willing, I would like some help on creating this list. I have already found a few, but I can't see any pattern that links them together besides c looks like its always a multiple of a prime number that is one more than a multiple of four.

• $$x^2+x^2 = 2x^2$$
• $$x^2+(7x)^2=2(5x)^2$$
• $$(7x)^2+(17x)^2=2(13x)^2$$
• $$(7x)^2+(23x)^2=2(17x)^2$$
• $$x^2+(41x)^2=2(29x)^2$$

I don't think that you can find a list of all equations, because of how the equation works. though I have found a method of calculating the amount of solutions for a given $$c$$ assuming $$a$$,$$b$$, and $$c$$ are natural numbers and are not equal to the same value.

1. choose $$c$$,
2. factor into prime numbers,
3. ignore 2 and all primes one less than a multiple of 4,
4. put the coefficients of the remaining primes in the equation $$\frac{(2x_1+1)(2x_2+1)(2x_3+1)...-1}{2}$$

ex: $$c=3900$$
$$3900=2^2*3^1*5^2*13^1$$
ignore $$2,3$$
coefficient of $$5^2$$ and $$13^1$$
the answer is $$\frac{(2*2+1)(2*1+1)-1}{2}=7$$