In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation
of Fermat's Last Theorem for odd prime
Specifically, Sophie Germain proved that at least one of the numbers
must be divisible by
if an auxiliary prime
can be found such that two conditions are satisfied: Conversely, the first case of Fermat's Last Theorem (the case in which
) must hold for every prime
for which even one auxiliary prime can be found.
Germain identified such an auxiliary prime
The theorem and its application to primes
less than 100 were attributed to Germain by Adrien-Marie Legendre in 1823.
[1] While the auxiliary prime
for which the violation of the Fermat Theorem would occur and most likely the conjecture is true that for given
the auxiliary prime may be arbitrarily large similarly to the Mersenne primes she most likely proved the theorem in the general case by her considerations by infinite ascent because then at least one of the numbers
must be arbitrarily large if divisible by infinite number of divisors and so all by the equality then they do not exist.