Hello,

given $\displaystyle n$ any

*composite* number, and $\displaystyle x^2 \equiv y \pmod{n}$, $\displaystyle x \neq y$, am I guaranteed of finding a "partitioning" of $\displaystyle y$ such that $\displaystyle x^2 - y = x^2 + bx + c$ for any $\displaystyle b, c \in \mathbb{N}$

*satisfying the condition* $\displaystyle b^2 - 4c$ a perfect square? And if yes, are there some predisposed values of $\displaystyle x$ for which it works? And (sorry) if the previous question has a negative answer, is there an easy way of finding the values of $\displaystyle b$ and $\displaystyle c$ that produce a perfect square given $\displaystyle x$ and $\displaystyle y$ ?

For instance, if $\displaystyle n = 77$, we choose say $\displaystyle x = 19$, and therefore $\displaystyle y = 53$, so we get $\displaystyle 19^2 - 53$. This can be written as $\displaystyle 19^2 - 19 \times 2 - 15$, and $\displaystyle 2^2 - 4 \times (-15) = 64$ which is a perfect square.

*Additional information* :

- the prime factorization of $\displaystyle n$ will not be used.

(for the curious, this leads to a factorization of $\displaystyle n$ - the perfect square requirement is necessary otherwise one needs to take square roots modulo $\displaystyle n$)

Thank you