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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.