Germain's Theory germain is a contemporary of French female mathematicians Cauchy and Legendre. She communicated with them, and her theorem was popular because of the admiration of great men. We are going to explain how beautiful and clever her result is. Its core is the following theorem: This theorem makes P and Q different odd prime numbers and satisfies the following conditions: 1. For any integer k, kpp (mod q); 2. If x, y and z are integers, and xp+yp+zp≡p(mod q) is q divisible by x, y or z, then the first case of FLT holds for the exponent p, and the theorem is proved by Barrow-Abel relation. Germain's famous theorem is as follows (1823): Theorem. Then the first case p, FLT is established. The proof of the theorem is very wonderful, which only involves the calculation of Legendre symbols and Fermat's theorem in elementary number theory. We wrote it for readers to appreciate. It is very easy to prove that prime numbers p and q=2p+ 1 satisfy the condition of theorem 1. If p≡ap(mod q), it is impossible to get p ≡ 1 (mod q). Secondly, suppose xp+yp+zp≡0(mod q) and qxyz ... Because p = (q- 1)/2, and Fermat's last theorem is applied, then XP ≡1(mod q) YP ≡1. □ Note that the quadratic congruence of prime modulus p can be written as x2≠α(mod p). When it has a solution, we say that α is the square residue of p; Otherwise, when it has no solution, we say that α is the square non-residue of p, and we have such a good result that we immediately produce several generalizations. In fact, in the same way of thinking, but with a slightly more detailed analysis method, Legendre proved the following results: Theorem 3 holds for the odd prime index p in the first case of FLT, as long as one of the following numbers is also prime: 4p+ 1, 8p+655. 10p+ 1， 14p+ 1， 16p+ 1。 Because of this theorem, the results of germain and Legendre contain all the prime numbers of P < 100, thus confirming the first case of these prime numbers. Although, moreover, it was proved as early as 1823. After all, this method has its limitations. Its difficulty lies in giving the prime number p, so 2kp+ 1 is also a prime number. When k is a large number. Secondly, this method does not work in the second case. More modern results are given by Crasner (1940) and Dines (195 1). Like Dines' method, Crasner's method is not completely simple. In fact, they used the results of modern algebra. This is Crasner's theorem. Theorem 4 assumes that p is an odd prime number. 2.3h3.3h/2 < 2hp+ 1； 4.22H 1 (model q). Then for p, the first case of FLT holds. 195 1 year Dines proves: Theorem 5 If P is an odd prime number, H is an integer, not a multiple of 3, h≤55, and q=2hp+ 1 is a prime number,