Who was Pierre de Fermat?

Anyone who has studied math has probably stumbled upon Pierre de Fermat. It could be his Two Squares Theorem, his Little Theorem, or even his famously unproven Last Theorem. His contributions to the study of mathematics have been countless, from fields from Calculus to Probabilities, some of which are being studied today! How did a lawyer from France make such groundbreaking contributions in math? Read more to find out!

The Life of a Mathematician

Pierre Fermat was born in either 1607 or 1608 in southern France. His father was a wealthy businessman, and his mother was from a high class family. Although little detail is known about his early education, there is enough evidence to suggest that it was a very effective education - Fermat was fluent in classical Greek, Latin, Spanish, Italian, and Occitan! 

From a young age, Fermat had an interest in mathematics, and this interest blossomed when he turned 19. It was at this age that he began a career as a lawyer in Bordeaux. While there, he became friends with Etienne d'Espagnet, who inherited several mathematical books that Fermat was able to borrow. Shortly after he began practicing law, his father passed away, leaving Fermat with a substantial inheritance. In 1630, at the age of 23, Fermat paid a large amount of money to gain a senior position in the High Court of Toulouse. This was a lifelong position of nobility. Because of the nobility associated with the position, Fermat was known as Pierre de Fermat from this time until his death. 

Fermat's Theorems and Conjectures

Much of Fermat's math was done in letters to companions. Although he claimed that he had proven all of his theorems and conjectures, very few of his proofs survived. However, over the years, mathematicians have been able to prove much of what he wrote. These theorems are just a small selection of the vast contributions Fermat made to mathematics! 

His two square theorem deals with prime numbers. The theorem states that any prime number will be the sum of two squares if and only if the prime number has a remainder of 1 when divided by 4 (or if the prime number is 2). We can see this is true with some small prime numbers: 5 and 13 each have a remainder of 1 when divided by 4, and they are each the sum of two squares (5 = 4 + 1 and 13 = 9 + 4). However, looking at 3 and 7, we can see that neither of them equal the sum of two squares. This holds true with all prime numbers!

Fermat's little theorem is simple in scope, but is used today in coding! The theorem states that if there are two integers a and p such that p is prime and not a factor of a, then a raised to the p-1 power divided by p will always have a remainder of 1. To show this, take p=3 and a=8. 8 * 8 = 64, and 64/3 = 21 R 1. This will work for all prime numbers, which allows mathematicians to use this theorem to check if large numbers are prime!

The last theorem we'll discuss is, fittingly, Fermat's Last Theorem. Fermat passed away before he was able to prove this, and for over three hundred years, mathematicians struggled to solve it - it was solved in 1995 by Andrew Wiles! The theorem states that for positive integers x, y, and z, the equation

x^n + y^n = z^n 

will never be true for n>2. Hundreds of years passed before Wiles was able to prove it, with several notable mathematicians who specialized in number theory all trying solve it! The Guinness Book of World Records had even named it the world's most difficult math problem, owing to the number of unsuccessful proofs!