# RSA use prime p as public exponent

I've got two 1024 bits prime $$p$$,$$q$$,and $$n$$ = $$p$$ * $$q$$. now I know the result of $$c^{p} \quad mod \quad n = x$$,also the value of c is given, I wonder if it is possible to factorize $$n$$.

From Fermat's little theorem we know $$a^p \equiv a \pmod{p}\,.$$ Applying this to the present problem, $$c^p \equiv c \equiv x \pmod{p}$$, and thus $$p = \gcd(x - c, n)$$.