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 )
Your conditions boil down to a system of equations.
This last equation is asking if is ever a square.
From this we can get
Thus we require to be square since we're wanting .
So let's try an example:
Let , .
We then get , so now we need to see if is ever square.
We now must solve ,or
You could then find the prime factorization of and test the finite amount of cases to solve for and ...
Or you could just believe me when I say there are no solutions in .
This means there is no such that for this particular example.
Thus we aren't always guaranteed there to be a square that satisfies your conditions.