Why Did Sophie Germain Discover Germain Primes?

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Sophie Germain discovered Germain primes while working on Fermat's Last Theorem, proving its first case for primes where the exponent divides one of the bases. Her work demonstrated that this theorem holds true for every Sophie Germain prime, as well as for other primes up to 100. Germain primes have applications in number theory and cryptography, highlighting their significance in mathematical research. The discussion emphasizes the importance of Germain's contributions to the understanding of prime numbers and their properties. Overall, her discoveries have had a lasting impact on the field of mathematics.
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Why did Germain come up with her Germain primes? I am intrigued to know why Sophie came across these primes. Do they have any applications?
 
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According to wikipedia:

Germain proved that the first case of Fermat's Last Theorem, in which the exponent divides one of the bases, is true for every Sophie Germain prime, and she used similar arguments to prove the same for all other primes up to 100. For details see Edwards, Harold M. (2000), Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory

http://en.wikipedia.org/wiki/Sophie_Germain_prime
 

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