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A number $c$ is given. We need to find a number $0<k<c$ such that

$c^2 - k^2$ is a perfect square. (if it is possible)

$c$ and $k$ can be any positive integer.

What I tried is-

  1. I iterated for all the values $[1,c-1]$ and stopped at the first $k$ which satisfies the condition.

  2. The Euclid's method to find a Pythagorean triple which takes O(c0.5) run time. We do not actually find a solution. But, we just need to check if it is possible or not.

Can the solution be better than this? What I am looking for is a more efficient solution. Maybe some algebraic formula or proof.

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There is a method based on the observation that according to Euclid's solution, if $c$ is the hypothenuse of an integral right triangle then there are coprime integers, $n$ and $m$ and an integer $t$ such that

$$ c = t^2(m^2+n^2) $$

So the problem will be solved if we find a way to write an integer $c$ as sum of two squares. The algorithm works as follows:

  1. Factor $c = t^2p_1p_2\dots p_s$. where the $p_i$ are distinct primes and $t$ is an integer. There will be solutions to the problem if and only if all the $p_i$ are $\equiv 1 \pmod{4}$. This is the most time consuming part but it can be done much faster than $O(\sqrt{c})$ using modern factoring algorithms.

  2. For each $p_j$ use Shank's algorithm (or other) to find an square root of $-1 \pmod{p}$. Although Shanks' is a probabilistic algorithm it is very fast. There is an algorithm (Schoof's algorithm) which finds square roots modulo prime in deterministic polynomial time but it is very complicated.

  3. Using the result of 2, Use Cornacchia's algorithm to find integers $a_j$ and $b_j$ with $$ a_j^2+ b_j^2 = p_j$$ see also the answers to this question

  4. Use the classical formula for the product of two sums of two squares $$ (a^2+b^2) (c^2+d^2) = (ac+bd)^2 + (ad-bc)^2$$ recursively to obtain a solution of $c/t^2 = A^2 + B^2$, then set $k=tA$ and you are done. You can also use complez arithmetic to do this in a simpler but equivalent way: $$ A + B i = (a_1+ib_1)(a_2+ib_2)\dots(a_s+ib_s) $$

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