Hello,

given

any

*composite* number, and

,

, am I guaranteed of finding a "partitioning" of

such that

for any

*satisfying the condition* a perfect square? And if yes, are there some predisposed values of

for which it works? And (sorry) if the previous question has a negative answer, is there an easy way of finding the values of

and

that produce a perfect square given

and

?

For instance, if

, we choose say

, and therefore

, so we get

. This can be written as

, and

which is a perfect square.

*Additional information* :

- the prime factorization of

will not be used.

(for the curious, this leads to a factorization of

- the perfect square requirement is necessary otherwise one needs to take square roots modulo

)

Thank you