Fermat

The life of Pierre de Fermat

Timeline c. 1607Born at Beaumont-de-Lomagne, in the south of France. The exact year is disputed. 1626Graduates in law from the University of Orléans. 1629–1631In Bordeaux, studies the work of Viète and reconstructs lost books of Greek geometry. 1631Becomes a councillor in the Parlement of Toulouse. Gains the right to use “de” in his name. 1636Begins corresponding with Marin Mersenne. He already has his analytic geometry and his method of maxima and minima. 1637Around this year, writes in the margin of Diophantus’s Arithmetica the note of the “last theorem”. 1640Announces the little theorem and the conjecture about the numbers $2^{2^n}+1$. 1652Nearly dies of plague. He is even reported dead. 1654Exchanges letters with Blaise Pascal and, with him, founds probability theory. 1662Derives the law of refraction from the principle of least time. 1665Dies on 12 January in Castres. 1670His son Samuel publishes Diophantus’s Arithmetica with his father’s margin notes. ◀ Previous Follow the journey Next ▶ c. 1607 Beaumont-de-LomagneHe is born into a family of leather merchants. The exact year is still debated. 1626 OrléansHe graduates in law from the University of Orléans. 1629–1631 BordeauxHe studies the work of Viète and reconstructs lost books of Greek geometry. He already has his method of maxima and minima. 1631–1665 ToulouseA councillor in the Parlement. In his spare time he writes to Mersenne and Pascal and notes the “last theorem” in the margin of Diophantus. 1665 CastresHe dies on 12 January, while serving at the court in Castres. Enable JavaScript to see the map. The stages are listed below. Fermat’s life in south-western France, between the law and mathematics. Map: © OpenStreetMap contributors

October 11, 2026 Read

Fermat and analytic geometry: a curve in an equation

For almost two thousand years, geometry and algebra lived apart. Greek geometry worked with ruler, compass and figures. Algebra, which reached Europe through the Arabs, worked with equations. At the start of the 17th century, two people had, at the same time and without consulting each other, the idea of joining them: René Descartes and Pierre de Fermat. A sentence that created a field Around 1636 Fermat wrote a short Latin text, Introduction to plane and solid loci. “Locus” was the Greek name for a set of points satisfying a condition, like the circle, the locus of points at a fixed distance from the centre. Right at the start, Fermat wrote:

October 11, 2026 Read

Maxima and minima: the rectangle with the largest area

You have 24 metres of fencing and want to enclose a rectangular plot with the largest possible area. Which dimensions should you choose? A long thin rectangle wastes fencing; a square seems better. But how do you prove it? Fermat invented a method for questions like this around 1636, more than thirty years before Newton and almost fifty before Leibniz published calculus. Fermat’s problem Fermat’s first example was exactly this: divide a segment of length $b$ into two parts, $x$ and $b-x$, so that the product $x(b-x)$ is as large as possible. That is our plot: with 24 m of fencing, two adjacent sides add up to $12$, and the area is

October 11, 2026 Read

Fermat's little theorem: powers and remainders

Take the number 2 and raise it to larger and larger powers. Divide each result by 7 and note the remainder: Power $2^1$ $2^2$ $2^3$ $2^4$ $2^5$ $2^6$ Value 2 4 8 16 32 64 Remainder mod 7 2 4 1 2 4 1 At $2^6$ the remainder is 1. Now try 3 and the prime 5: $3^4=81$, which leaves remainder 1 when divided by 5. And 10 and the prime 13: $10^{12}=1{,}000{,}000{,}000{,}000$, which also leaves remainder 1 when divided by 13.

October 11, 2026 Read

Fermat numbers: a conjecture and a counterexample

Mathematicians have always dreamed of a formula that produces only prime numbers. In 1640 Fermat thought he had found one. Five primes in a row Fermat studied numbers of the form $$F_n=2^{2^n}+1.$$The exponent is a power of 2: $1,2,4,8,16,\ldots$ Computing the first few: $$F_0=3,\quad F_1=5,\quad F_2=17,\quad F_3=257,\quad F_4=65{,}537.$$All five are prime. Checking $65{,}537$ by hand, trying divisors up to its square root, was already hard work. Fermat wrote to Frénicle de Bessy that he was convinced all numbers of this form were prime, although he admitted he had no proof.

October 11, 2026 Read

Fermat and Pascal: fairly dividing an interrupted game

Two players bet the same amount on a game of several rounds: whoever first wins a certain number of rounds takes everything. Midway through, the game has to be stopped. How should the money be divided fairly? This is the problem of points. It looks simple, but it defeated mathematicians for more than 150 years. The wrong answer In 1494 the Italian friar Luca Pacioli proposed dividing the prize in proportion to the rounds already won. If A has won 2 rounds and B has won 1, A takes $\frac23$ of the prize.

October 11, 2026 Read

Fermat's principle: light takes the fastest path

A lifeguard is on the sand and sees someone drowning in the sea, further out and off to one side. Which path should the lifeguard take? The straight line is the shortest path but not the fastest: running on sand is much quicker than swimming. It pays to run a bit further along the beach so as to swim less. Fermat realized that light makes exactly this choice. Reflection: the shortest path The oldest idea concerns mirrors. In the 1st century, Heron of Alexandria had already noticed that light, when reflected, follows the shortest path between two points by way of the mirror.

October 11, 2026 Read

Infinite descent: a smaller solution leads to a contradiction

Imagine a staircase where each step has a positive whole number. You can go down: 10, 7, 4, 2, 1… but at some point the staircase ends. There is no infinite sequence of ever-smaller positive integers. It seems obvious. Fermat turned this obvious fact into one of the most powerful tools in number theory: infinite descent. How it works To prove that an equation has no solution in positive integers:

October 11, 2026 Read

Fermat's last theorem: from the margin to Wiles's proof

Everyone knows Pythagoras’s theorem: $3^2+4^2=5^2$. There are infinitely many triples of whole numbers like this, such as $5,12,13$ and $8,15,17$. What about cubes? Are there positive integers with $a^3+b^3=c^3$? And with fourth powers? The note in the margin The page of problem II.8 in the 1670 edition of Diophantus’s Arithmetica. Just below the problem comes the “Observatio Domini Petri de Fermat”: the margin note, printed by Fermat’s son.Diophantus, ed. Samuel de Fermat (1670) · Wikimedia Commons · public domain Around 1637, reading Diophantus’s problem about splitting a square into two squares, Fermat wrote in the margin of the book, in Latin:

October 11, 2026 Read

Fermat's legacy

The father of number theory Title page of the 1670 edition of Diophantus’s Arithmetica, printed in Toulouse “with the observations of P. de Fermat”. It was through this book, published by his son five years after his death, that the margin notes reached Euler.Bernard Bosc, Toulouse (1670) · Wikimedia Commons · public domain Before Fermat, problems about whole numbers were seen as riddles, Diophantus’s puzzles. Fermat turned them into an area of mathematics, with deep questions and methods of its own. For almost a century hardly anyone followed him. It was Euler, from 1730 onwards, who read Fermat’s notes, proved many of them and made number theory a central branch of mathematics. Then came Lagrange, Legendre and Gauss.

October 11, 2026 Read

Image credits

Historical portrait The portrait attributed to Rolland Lefebvre comes from Wikimedia Commons and is in the public domain. We use the thumbnail provided by Commons, without editing, and the caption keeps the attribution given by the collection. Source page. Public domain mark. Blog figures The cover and the refraction sketch are original illustrations. The interactive figures for remainders of powers, the area of rectangles and the division of the interrupted game were created for this trail. They are teaching aids and do not reproduce Fermat’s manuscripts. The division figure, for example, assumes independent rounds and players with equal chances.

October 11, 2026 Read