Sophie Germain discovered her vocation for mathematics as a teenager, through her father’s books. The family disapproved of such an “unseemly” occupation for a girl from a family in 18th-century Paris, but she persevered and eventually achieved a reputation among the finest mathematicians of her time.
Reading the “Essay on the Theory of Numbers”, published by Adrien-Marie Legendre in 1798, and the “Arithmetic Investigations” (“Disquisitiones Arithmeticae”), which Carl-Friedrich Gauss wrote that same year and published in 1801, awoke his taste for number theory, which would be his main research topic.
His best-known work concerns Fermat’s theorem, according to which the equation xno+yno=zno has no integer solutions when the exponent n is greater than 2. The known results dealt with specific values ​​of the exponent: n=4 (Fermat, 1670), n=3 (Euler, 1770) and n=5 (Legendre and Dirichlet, 1825 ).
Germain was the first to treat a whole family of exponents: she proved that if n satisfies certain conditions —which hold for all integers less than 100—then any solution of the equation must be such that some of the numbers x, y, or z is a multiple of n (first case of Fermat’s theorem). In fact, this was the first step in an ambitious plan to prove the general case of the theorem. It ended up not working, but Germain’s pioneering spirit remains impressive.
The conditions of Germain’s theorem are automatically satisfied if the exponent n is a “prime of Germain”, that is, a prime number such that 2n+1 is also prime. The list of Germain primes starts with 2, 3, 5, 11, 23, 29, 41, … An intriguing question is how many there are: they are believed to be in infinite quantity and that there are at least N/(log N)two Germain primes smaller than any given integer N. But no one has yet been able to prove these facts.
Numbers of the form 2n+1, with n being a Germain prime, are called “safe primes”, due to a practical application that she could never have foreseen.
The main current methods of encryption are based on the fact that, given a product pq of two large primes, it is difficult to identify the factors p and q. But this depends on the choice of primes: for example, if p is such that p–1 can be factored into small primes, it is not so difficult to break the encryption. One way to avoid this risk is to use p and q that are safe primes.
