Finding Pythagorean triplet given the hypotenuse

Question

I have a number $c$ which is an integer and can be even or odd. It is the hypotenuse of a right angled triangle. How can I find integers $a,b$ such that

$$ a^2 + b^2 = c^2 $$

What would be the complexity of the calculation?

Answer 1

You may go by trial and error, or through the following lines. Consider the factorization of $c^2$ and separate its members according to them being $4^n$, $p^{2m}$ with $p\equiv 1\pmod{4}$ and $q^{2h}$ with $q\equiv 3\pmod{4}$. Every prime $p\equiv 1\pmod{4}$ splits in $\mathbb{Z}[i]$ and so it can be represented as a sum of two squares in a essentially unique way, $p=a_p^2+b_p^2$. That representation can be recovered with a finite descent that exploits Lagrange's identity: $$ (a_1^2+b_1^2)\cdot(a_2^2+b_2^2) = (a_1 a_2+b_1 b_2)^2 + (a_1 b_2-a_2 b_1)^2. $$ In order to compute $a_p$ and $b_p$ from $p$, we need about $\log(p)$ steps. Have a look at the second part of my answer to this question in order to understand how the descent really works.

Given the representations of every prime $p\equiv 1\pmod{4}$ dividing $c$, we may also recover, through Lagrange's identity, every representation of $c^2$ as a sum of two squares.

Answer 2

You can find one or more triples, given only $C$ if $GCD(A,B,C)=2$ or a perfect square, which includes all primitives and such as $(27,36,45)$. You cannot find those that are otherwise, such as $3,5,6,7,8,10$ times a primitive, e.g. $(45,24,51)=3*(15,8,17)$.

We solve $C=m^2+n^2$ for $n$, and look for positive integers

$$n=\sqrt{C-m^2}\text{ where }{\lceil \sqrt{\frac{C}{2}}\space\rceil\le m\le \lfloor\sqrt{C}\rfloor}$$.

If we want to find one or more triples for $C=145$, then we have $$m_{min}=\lceil \sqrt{\frac{145}{2}}\space\rceil=9\qquad m_{max}=\lfloor\sqrt{145}\rfloor=12$$

If we try $9\le m \le 12$, we find $n\in\mathbb{N}$ only for $\sqrt{145-9^2}=8$ and $\sqrt{145-12^2}=1$

$$f(9,8)=(17,144,145)\qquad f(12,1)=(143,24,145)$$

For $C=1105$, there are $4$ triples; for $C=29$, there is $1$ triple; for $C=31$, there are no triples. If $C=20$, we find $f(4,2)=(12,16,20)=2^2*f(2,1)=2^2*(3,4,5)$. Try the numbers and see for yourself.

Answer 3

c can be found as

$c=2 r^2 +s^2+2 r s$
$a=2r^2+2 r s$
$b=s^2+2 r s$

Edit Was using x^2+y^2=z^2 variables changed to a^2+b^2=c^2