# Pierre de Fermat: Contributions to Math & Accomplishments

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Pierre de Fermat was an amateur mathematician that contributed countless theorems and ideas to the field of mathematics. This lesson will explore some of these contributions and accomplishments.

## Pierre de Fermat

Pierre de Fermat was a French lawyer that was born around 1601-1607 in Beaumont de Lomagne, and died January 12, 1665, in Castres. We say his birth was around 1601-1607 because details of his early life are fairly sketchy, and it may be that he had an older brother, also named Pierre, that died young. Because of this, some say that Fermat's assumed birthday of August 17, 1601, is actually his brother's birthday, and Fermat was born sometime after that.

Despite his early life being a bit of a mystery, we do know that in his twenties, he attended the University of Orléans where he earned his degree in civil law. At this point, you may be wondering why we are talking about a lawyer in a math lesson, but this is what makes Fermat so fascinating.

You see, Fermat was a lawyer by trade and only studied math as a hobby. This may lead you to believe that his contributions to mathematics are few, but that is definitely not the case! Fermat was, and still is, one of the most well-known and brilliant mathematicians out there.

He never published any of his work, and rarely provided evidence of his claimed proofs for his many conjectures. Although he never published his works, he was in frequent contact with many other mathematicians through correspondence, so we have record of his contributions to the field of mathematics, which is good news, because there are many!

## Contributions to Mathematics

Fermat made contributions in many areas of mathematics, such as probability theory, analytic geometry, optics, and infinitesimal calculus. He was the inventor of modern number theory, and this was where a lot of his work was concentrated. For instance, Fermat discovered numbers of the form

• Fn = 22n + 1

These numbers are called Fermat numbers, and when a Fermat number is prime, it is called a Fermat prime. The only known and proven Fermat primes are for the cases when n = 0, 1, 2, 3, and 4. These numbers are extremely important to the study of prime numbers and Mersenne numbers, both of which are strongly studied in number theory.

Another area that Fermat made great advancements in is in the field of optics. Fermat's research in this area led to what is known as Fermat's principle, which states that 'the path between two points taken by a ray of light leaves the optical length stationary under variations in a family of nearby paths.' In other words, a ray of light always prefers the path that has other paths nearby along which the ray would take very nearly the exact same amount of time to traverse. From this principle, he deduced the laws of refraction and reflection.

## Fermat's Little Theorem and Fermat's Last Theorem

Of all of Fermat's mathematical contributions and accomplishments, he is best known for two of his theorems, and those are his Little Theorem and his Last Theorem.

Let's start with Fermat's Little Theorem. In mathematical notation, this theorem states that if p is a prime number, then for any integer a, where p does not divide a,

• ap-1 = 1(mod p)

In words, this says that if we have two numbers a and p, where p is a prime number and not a factor of a, then a multiplied by itself p - 1 times and then divided by p will always have a remainder of 1.

The Little Theorem isn't so little in all it has helped to create today. As an example, if you've ever used your credit card to complete an online transaction, you can thank Fermat and his Little Theorem for that transaction being secure. You see, the code for keeping our online credit card transactions secure highly revolves around this theorem.

A lesson on Pierre de Fermat wouldn't be complete without mention of Fermat's Last Theorem, which states that xn + yn = zn has no whole integer solutions for n > 2.

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