# Sophie Germain prime

If both*p*and 2

*p*+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 2*p*+1 divides the Mersenne number
M_{p}.

**See Also:** CunninghamChain

**Related pages** (outside of this work)

- Sophie Germain primes (records and theorems)
- Euler and Lagrange's theorem (proof of the theorem mentioned above)
- Math's Hidden Woman (about the woman Sophie Germain)

**References:**

- Agoh2000
Agoh, Takashi, "On Sophie Germain primes,"Tatra Mt. Math. Publ.,20(2000) 65--73. Number theory (Liptovský Ján, 1999).MR 1845446- CFJJK2006
Csajbók, T.,Farkas, G.,Járai, A.,Járai, Z.andKasza, J., "Report on the largest known Sophie Germain and twin primes,"Ann. Univ. Sci. Budapest. Sect. Comput.,26(2006) 181--183.MR 2388687- Dubner96
H. Dubner, "Large Sophie Germain primes,"Math. Comp.,65:213 (1996) 393--396.MR 96d:11008(Abstract available)- FG91
G. FeeandA. Granville, "The prime factors of Wendt's binomial circulant determinant,"Math. Comp.,57:196 (1991) 839--848.MR 92f:11183- JR2007
Jaroma, John H.andReddy, Kamaliya N., "Classical and alternative approaches to the Mersenne and Fermat numbers,"Amer. Math. Monthly,114:8 (2007) 677--687.MR 2354438- Peretti1987
Peretti, A., "The quantity of Sophie Germain primes less thanx,"Bull. Number Theory Related Topics,11:1-3 (1987) 81--92.MR 995537- Yates1987
Yates, Samuel,Sophie Germain primes. In "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991. pp. 882--886,MR 1146271

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