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Sophie Germain: cut down in her prime

This is the second in the series on mathematicians that I started with Pythagoras. Diversity is a big issue for society today … indeed, it has been a big issue for society for many a long year. The difference now is that we recognise it as an issue; at least, we think we do. A newish acronym in the area of gender diversity is STEM. This stands for science, technology, engineering and mathematics. There aren’t enough women in STEM and there aren’t enough women and girls doing STEM subjects at school and university. When I was at school we boys did woodwork and the girls did domestic science. You look back at that and you wonder (with the benefits of hindsight, of course) that it was a pretty silly distinction.

There are far fewer well-known female mathematicians than there are male mathematicians although this is changing. Perhaps you read my recent obituary of Maryam Mirzakhani in the The Beagle. Closer to home, but still very much alive is Nalini Joshi who is the first woman to be a Professor in the School of Mathematics and Statistics at the University of Sydney. Even so, as she says, when she is attending mathematical gatherings in a black suit she is sometimes mistaken for the hospitality staff. This is a sad state of affairs. But there is a ray of light. I have a wonderfully readable and slim volume called “Sex and Mathematics” by a woman called Clio Cresswell. I was delighted to discover that Dr Cresswell was one of Professor Joshi’s PhD students.

My standout female mathematician of the past is Sophie Germain (1776 – 1831). Perhaps this is because I have a daughter called Sophie; perhaps my daughter is called Sophie because of Sophie Germain. Who can say? You would guess from her name that Sophie Germain is French and you would be right.

Sophie Germain

Some writers say that of all the European countries, 18th and 19th century France displayed the most discriminatory attitude toward educated women. The view then in France was that mathematics was unsuitable for women. Yet French mathematicians dominated many fields of mathematics in the 18th and 19th centuries.

Sophie was the second of three daughters of a prosperous silk-merchant and her life was lived against the backdrop of the French Revolution. While women of her background were not encouraged to be mathematicians, they were required to have enough knowledge to hold a polite conversation. One of the books she read was “Sir Isaac Newton’s Philosophy Explained for the Use of Ladies”.

Its author, one Francesco Algarotti, thought that women were only interested in romance. He explains the law of gravitation and says that “this proportion in the distances of places is observed even in love. Thus after eight days’ absence one’s sense of longing becomes sixty-four times less than it was on the first day.” Not surprisingly, Sophie didn’t have much time for such nonsense.

Instead she read a life of Archimedes that she found in her father’s library. Archimedes is supposed to have been killed as a result of being so distracted by his geometry that he did not respond to a Roman soldier and was speared. Sophie reckoned that if mathematics was that engaging then it would be the thing for her. She started reading. Her parents removed her candles and confiscated her clothes in an effort to keep her from this unfeminine pursuit. When the École Polytechnique opened in 1794 Sophie was denied entry; of course, she was a woman. But she masqueraded as another student called, improbably, M le Blanc. In this way she got hold of the lecture notes and problems. Each week she submitted answers to the problems. The course supervisor was Joseph-Louis Lagrange who was then, and is still now, one of the greatest mathematicians. M le Blanc had been a pretty average student and Lagrange was mystified about how he could suddenly have become so capable. Sophie was outed but Lagrange, to his great credit, became her mentor.

Sophie became interested in Fermat’s Last Theorem. A couple of month’s ago, in these august pages, I wrote about Fermat’s Last Theorem. Sophie did more to advance the understanding of Fermat’s Last Theorem than any mathematician since Fermat formulated it in 1637 until the work of a later mathematician (Eduard Kummer (1810 – 1893)) in 1840. Sophie’s approach used a particular type of prime number. A prime number is a number that is divisible only by itself and one. Examples are 7, 13 and 101. She was interested in what are now called Sophie Germain Primes. A Sophie Germain Prime is a number p such that 2p + 1 is also prime. Examples are 5 (2p + 1 = 11, which is also prime) and 593 (2p + 1 = 1187, which is also prime). She was able to prove Germain’s Theorem which showed that Fermat’s Last Theorem was true for certain values related to Sophie Germain Primes.

Now, something called public key cryptography depends upon the properties of very large primes called safe primes; a Sophie Germain Prime is a safe prime. It is public key cryptography that keeps our data secure as it travels around the Internet. Internet data security requires very large primes … perhaps as large as the Sophie Germain Prime 1,846,389,521,368 + 11600.

The largest known Sophie Germain Prime is 2,618,163,402,417 × 21290000 − 1 which has 388,342 digits. No one knows how many Sophie Germain Primes there are; I suspect that there are an infinite number but no one has proved that. And, as we all know, a mathematician without a proof is like a door without a handle.

In the early years of the 19th century her interests switched from number theory to applied mathematics. If you place grains of sand on a thin metal plate and then “play” the plate with a violin bow, the grains of sand move about and eventually settle into patterns. The patterns are called Chladni patterns after the man that Sophie saw demonstrate the phenomenon. You can watch this on YouTube.

Napoleon authorised the creation of a special prize to explain why these patterns should form. This was the Prize of the Paris Academy of Sciences and Sophie went for it in 1811. She was not awarded the prize but neither did anyone else win it. She did not succeed two years later in 1813; her submission was “full of holes”. But she was successful in 1816. A comparable award today would be a Fields Medal or an Abel Prize; at the time French mathematics was the peak of the discipline.

In 1829 she found that she had breast cancer, the same brutal disease that took Maryam Mirzakhani earlier this month. She died in 1831. Her death certificate describes her as a property holder rather than a mathematician. She wasn’t the only female mathematician of her generation and perhaps she wasn’t the greatest; Maria Agnesi (1718-1799) was the first female to be appointed as a mathematics professor, Ada Lovelace (1815-1852) is renowned as the world’s first computer programmer and, a little later and possibly greater, Sofia Kovalevskaya (1850-1891), the first woman appointed to a full professorship in Northern Europe.

Yet Sophie Germain has always captured my imagination, possibly because she was a number theorist. For as someone - I cannot remember who - said; if mathematics is the queen of science then number theory is the queen of mathematics. She gave us Sophie Germain Primes and so contributed to the Internet we use today. And you can help look for Sophie Germain Primes through Prime Grid and the Twin Prime Search. These are two examples of how you can contribute to scientific research by allowing your computer to be used while it is connected to the Internet but otherwise lying idle. I use this to help look for Woodhall Primes. You can look those up yourself.

Further reading: Mathematics and Sex, Clio Cresswell, Allen & Unwin, 2003, ISBN 1 74114 159 1 (I think this reading is OK for younger people; in fact, it should probably be encouraged. Do not be put off by the chapter entitled “How much sex is too much?”. This chapter contains some really important mathematical insights into how you should choose your partner for life).

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