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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) If both p and 2p+1 are prime, then p is a Sophie Germain prime. The first few such primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, and 131. Around 1825 Sophie Germain proved that the first case of Fermat's last theorem is true for odd Germain primes. Soon after Legendre began to generalize this by showing the first case of FLT also holds for odd primes p such that kp+1 is prime, k=4, 8, 10, 14, and 16. In 1991 Fee and Granville extended this to k < 100, k not a multiple of three. Many similar results were also shown, but now that Fermat's Last Theorem has been proven by Wiles, they are of less interest. Euler and Lagrange proved the following about Sophie Germain primes: if p 3 (mod 4) and p > 3, then the prime 2p+1 divides the Mersenne number M_{p}.
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Chris K. Caldwell © 19992019 (all rights reserved)
