Beautiful Minds

Imagine that you are trying to find the remainder that you will arrive at when dividing a huge number by another number (the modulo): it would likely be a pain in the neck to brute force. For example, have you ever tried to calculate the remainder of 3^100/31? To a normal person, this seems like a rather insane task that would take a whole weekend; to Pierre de Fermat, however, it was a one-sentence trick that barely even required paper, let alone a calculator. 

Pierre de Fermat was anything but a normal mathematician in his era: living as a French mathematician in the 17th century, Fermat was a man with many original discoveries. 

Pierre de Fermat (1601–1665) was a French lawyer and magistrate who, despite pursuing mathematics only as a passionate hobby, is regarded as one of the 17th century’s most brilliant mathematicians and the founder of modern number theory. Operating in near-total isolation from other mathematicians (mathematics has since become more collaborative), he laid the foundations for probability theory and early calculus. Interestingly, Fermat was actually a lawyer before he was a mathematician! He graduated with a bachelor’s degree in civil law, and he was seen in his time as a hobbyist, amateur mathematician who worked on conjectures and problems in his spare time. 

One notable contribution that Fermat made to the world of math was a finding in integral calculus: Fermat was one of the mathematicians who helped generalize the antidifferentiation process by deriving the reverse power rule. Although we often take this for granted in the present (as this formula is taught to many high schoolers these days in AP Calculus BC and other Calculus classes), we have to contextualize his findings in his time period: back then, it was revolutionary to discover the purpose of doing the derivative process backward and realizing that it would yield the area under a curve. I always found it fascinating that the slope of a curve and the area under it are found using operations that are inverses of each other, and learning this fact in Calculus BC and extending the formula to find areas between curves felt borderline mind-bending at first. 

Another key discovery of Fermat’s is a theorem that is used to calculate remainders of large numbers, amusingly referred to as “Fermat’s Little Theorem”. It is a versatile and useful theorem that is applied to many problems, yet it is fundamentally simple. At its core, the theorem is a rule about how prime numbers behave when they are used as exponents. More specifically, the rule states that if you take any whole number and raise it to the power of a prime number, then subtract the original number, the result will always be an exact multiple of that prime. It is a fundamental law of numerical symmetry that Fermat discovered while playing with numbers in his spare time, being the type of mathematician that he was. If you are trying to find the remainder of a massive number raised to a huge power, Fermat’s Little Theorem allows you to collapse those giant exponents into tiny, manageable pieces. For example, if you are dividing by a prime number like 31, the theorem tells you that any number raised to the 30th power will always leave a remainder of 1. This greatly simplifies any remainder calculation by reducing complex expressions through concise operations, turning a day-long problem into a minute-long one. Although it has its limitations, such as not granting the actual quotient value itself (somewhat obviously, since this is something only a calculator can do quickly), it is a stroke of mathematical ingenuity for a theorem discovered in the 1600’s. 

Fermat’s theorem is useful both for quick problem-solving and detailed number theory proofs. In the competition math circuit, Fermat’s Little Theorem is a must-have tool. In my experience, although it is often not the only thing applied in order to reach a solution, it is a crucial step in several different kinds of problems in which the remainder or divisibility of a number must be quickly deduced without a calculator. Some problems present you with huge numbers that couldn’t even be approached otherwise. 

However, Fermat’s influence stretches far beyond the classroom or the math competition stage. You actually use his work every single time you buy something online or log into your email. Digital security relies on a concept that is simultaneously simple and nuanced: it is very easy to multiply two massive prime numbers together, but it is incredibly difficult for a computer to undo that work and find the original factors. Fermat’s mathematical rules provide the key that allows your computer to encrypt your credit card information so that only the bank’s computer can unlock it. Without this 17th-century lawyer’s hobby, the private data we send across the internet would be vulnerable to anyone with a fast enough processor.

In the end, Pierre de Fermat’s legacy is about his specific way of seeing the world. Fermat wasn’t chasing a paycheck for his math; he was chasing the joy of the puzzle. Lawyer by day and a mathematician by night, he spent his days in the courtroom and his nights uncovering the hidden patterns of the universe, often leaving his most brilliant thoughts as mere scribbles in the margins of his books. Whether you are a student solving a high school math problem or a consumer securing a bank transfer, you are standing on the shoulders of a 17th-century French lawyer who saw math as a game. Next time you see a giant exponent, don’t reach for the calculator; rather, remember Fermat and his Little Theorem.

Bibliography

O’Connor, John J., and Edmund F. Robertson. “Pierre Fermat (1601–1665).” MacTutor History of Mathematics, University of St Andrews, Dec. 1996, mathshistory.st-andrews.ac.uk/Biographies/Fermat/. Accessed 14 Apr. 2026.

Mahoney, Michael Sean. “Pierre de Fermat.” Encyclopedia Britannica, 11 Jan. 2024, www.britannica.com/biography/Pierre-de-Fermat. Accessed 14 Apr. 2026.

Pellegrino, Dana. “Pierre de Fermat.” Rutgers Math, Rutgers University, sites.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html. Accessed 14 Apr. 2026.

“Pierre de Fermat.” Wikipedia, Wikimedia Foundation, 4 Apr. 2026, en.wikipedia.org/wiki/Pierre_de_Fermat. Accessed 14 Apr. 2026.