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