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 )