As a woman, Germain had many obstacles to overcome before her contributions to science were accepted, not only in the field of mathematics, but also in acoustics and the study of elasticity.

Germain was born in Paris, into a wealthy family. Her father was a merchant and later became a Director of the (national) Bank of France. They were also a family strongly interested in liberal reforms - a common theme during the French and American Revolutions taking place at the time. Her family home was a meeting place at which she was introduced to the topics of the day.

As was common at this time, women were not supposed to be interested- let alone be active - in studying subjects such as mathematics. But Germain was an exception. Against her family’s wishes, the young Germain would spend hours in her father’s library at night studying, when the rest of the household was asleep. Her parents relented in the face of stubbornness, and they accepted their daughter’s wishes.

Germain was to face further obstacles still, when trying to enrol in the École Polytechnique, Paris in 1794, as women were simply not permitted to attend.

Not discouraged, she obtained the lecture notes from other students and continued to teach herself. During this period Germain became fascinated with the work of Lagrange, a professor there, and submitted a paper to him on Analysis, under a man’s name.

Having gained the attention of Lagrange, who was so impressed with its originality that he began a search for its author. Once Lagrange had accepted her deception he took it upon himself to become Germain’s mentor.

The support that Lagrange provided was all that Sophie needed. With this new found encouragement she began submitting papers into competitions.

The most notable was sponsored by the French Academy of Science, the competition ran from 1808 to 1816 and was based on a previous study by a German physicist on the subject of elastic surface vibrations. It took Germain three attempts before she was awarded this accolade!

But why was Sophie interested in primes? Simple, she thought that it might lead to a proof of the notorious Fermat’s Last Theorem.

Around 1825, Sophie Germain proved that the first case of Fermat’s Last Theorem is true for her primes, i.e. if *p* is a Sophie Germain prime, then there do not exist integers *x*, *y*, and *z* different from 0 and not multiples of *p* such that x^p+y^p=z^p.

This was a breakthrough, but little progress has subsequently happened as a consequence of this result or the idea of such primes.

Germain was the first woman to be allowed to attend the conferences set up by the Academy of Science, yet once again her research was over looked for many years to come. It was not until the intervention of Gauss in 1831 that Germain was awarded an honorary degree from the University of Göttingen. Alas this came too late for her to receive it - Germain died of cancer, having battled the disease for many years, at the age of 55.

A prime p is said to be a Sophie Germain prime if both p and 2p+1 are prime. The first few Sophie Germain primes are:

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, \ldots

Large Sophie Germain primes include:

- 92305.2^{16998}+1, found in 1998. This number has 5117 digits;
- 109433307.2^{66452}-1 found in 2001. This number has 20013 digits.

It is still not known if there are an infinite number of Sophie Germain primes. However, if there are, then we now know that the number of such primes less than n, denoted by \lambda_G(n) is given by:

{\lambda_G(n)}\approx {n\over(\log n)^2}\ (\approx\ \mbox{means approximately)}

Using this approximation, we can compare the estimate with the actual number:

N | actual | estimate |

1,000 | 37 | 39 |

100,000 | 1171 | 1166 |

10,000,000 | 56032 | 56128 |

100,000,000 | 423140 | 423295 |

1,000,000,000 | 3308859 | 3307888 |

10,000,000,000 | 26569515 | 26568824 |