Sophie Germain's theorem
In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation x p + y p = z p {\displaystyle x^{p}+y^{p}=z^{p}} of Fermat's Last Theorem for odd prime p {\displaystyle p} . == Formal statement == Specifically, Sophie Germain proved that at least one of the numbers x {\displaystyle x} , y {\displaystyle y} , z {\displaystyle z} must be divisible by p 2 {\displaystyle p^{2}} if an auxiliary prime q {\displaystyle q} can be found such that two conditions are satisfied: No two nonzero p t h {\displaystyle p^{\mathrm {th} }} powers differ by one modulo q {\displaystyle q} ; and p {\displaystyle p} is itself not a p t h {\displaystyle p^{\mathrm {th} }} power modulo q {\displaystyle q} .