Find the largest possible sum m+n for positive integers m, n \leq 100 such that m+1 \equiv 3(\bmod~ 4) and there exists a prime number p and nonnegative integer a such \frac{m^{2^{n}-1}-1}{m-1}=m^{n}+p^{a}.
Find the largest possible sum m+n for positive integers m, n \leq 100 such that m+1 \equiv 3(\bmod~ 4) and there exists a prime number p and nonnegative integer a such \frac{m^{2^{n}-1}-1}{m-1}=m^{n}+p^{a}.