Sophie Germain: mathématicienne extraordinaire - Scripps College

What then 65
This means that if P is a prime,
(aP ) P + Y P + Z P ≡ aP + Y + Z (mod P )
≡ 0 + Y + Z (mod P )
≡ Y + Z (mod P )
≡ 0 (mod P ),
so Y ≡ −Z (mod P ).
In fact, by Chinese Remainder Theorem, if P is a product of different
primes, we get exactly the same results.
Chinese Remainder Theorem.
Let b, c, m, and n be integers such that m and
n are relatively prime. Then the simultaneous congruences
x ≡ b (mod m) and x ≡ c (mod n)
have exactly one solution with 0 ≤ x < mn.
In particular, if x ≡ b (mod m) and x ≡ b (mod n), then x ≡ b (mod mn).
Since P is a product of different primes p i and aP ≡ 0 (mod p i ) for all i,
then aP ≡ 0 (mod P ) and Y ≡ −Z (mod P ).
Therefore, if X P + Y P + Z P = 0 has non-trivial integer solutions, x, y,
and z when P is a product of unique primes, then x is divisible by P and
y = bP − z for some integer b.
This typical number theoretical approach to Fermat’s Last Theorem is
possible due to the consequences of Sophie Germain’s Theorem. Her ap-