1601 – 1665 • Prince of Amateurs
Lawyer by profession, mathematician by passion — Fermat founded modern number theory, co-founded probability theory, and left behind the most famous unsolved problem in mathematics.
Pierre de Fermat was born in late 1601 (traditionally August 17, though some sources suggest earlier) in Beaumont-de-Lomagne, near Toulouse, in southern France. His father, Dominique Fermat, was a wealthy leather merchant and second consul of the town.
Fermat studied law, probably at the University of Toulouse and the University of Orleans, where he earned his degree in civil law. He was fluent in French, Latin, Greek, Italian, and Spanish.
Mathematics was never Fermat's profession. Throughout his life, he was a magistrat — a judicial official in the Parlement of Toulouse. Mathematics was his private passion, pursued in the margins of legal duties and, famously, in the margins of books.
Fermat spent his entire career in the legal system of Toulouse, rising to the position of conseiller au Parlement. His judicial duties actually benefited his mathematical work: as a judge, he was discouraged from socializing, leaving ample time for research.
He communicated his discoveries through letters to Mersenne, Pascal, Huygens, Carcavi, and others, but rarely published formally. His son Samuel published his mathematical correspondence and marginalia posthumously in 1679.
Fermat's habit of stating results without proof (often claiming the proof was too long for the margin) has tantalized and frustrated mathematicians for centuries. Most of his claims were eventually proved correct.
Seventeenth-century European scholars communicated through elaborate networks of correspondence. Marin Mersenne in Paris served as the central hub, forwarding letters and challenges between Fermat, Descartes, Pascal, and dozens of others.
Fermat's number theory was inspired by his reading of Diophantus' Arithmetica in Bachet's 1621 Latin translation. His marginal notes in this book became the source of many famous conjectures, including the Last Theorem.
The early 1600s saw many mathematicians developing proto-calculus techniques: Cavalieri's indivisibles, Descartes' normals, Fermat's adequality. These disparate methods would be unified by Newton and Leibniz within a generation.
"I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
— Pierre de Fermat, marginal note in Arithmetica (c. 1637)Fermat's most powerful proof technique: assume a solution exists and derive a smaller solution, creating an impossible infinite descent.
Fermat used this to prove the case n = 4 of his Last Theorem: x⁴ + y⁴ = z⁴ has no positive integer solutions. The general case for all n ≥ 3 was proved by Andrew Wiles in 1995.
If p is prime and a is not divisible by p, then:
ap−1 ≡ 1 (mod p)
This is the foundation of modern primality testing (Miller-Rabin) and public-key cryptography (RSA). Fermat stated it in a letter to Frenicle de Bessy in 1640 without proof; Euler proved it in 1736.
An odd prime p can be written as the sum of two squares if and only if p ≡ 1 (mod 4). Fermat claimed a proof by infinite descent; Euler finally proved it in 1749.
Fermat conjectured that all numbers of the form F_n = 2^(2^n) + 1 are prime. The first five are indeed prime (3, 5, 17, 257, 65537), but Euler showed F_5 = 4294967297 = 641 × 6700417 is composite. No further Fermat primes are known.
Fermat claimed every positive integer is the sum of at most 3 triangular numbers, 4 squares, 5 pentagonal numbers, etc. The four-square theorem was proved by Lagrange (1770); Cauchy proved the general case in 1813.
In 1654, the Chevalier de Mere posed the "Problem of Points" to Pascal, who wrote to Fermat. Their exchange founded probability theory.
Fermat solved the Problem of Points by exhaustively listing all possible outcomes of the remaining rounds. He computed the fraction of outcomes favorable to each player.
His approach was essentially: enumerate the sample space, count favorable events, divide. This is the classical definition of probability that Laplace would later formalize.
Fermat's combinatorial method generalized easily to any number of remaining rounds, making it more systematic than Pascal's recursive approach (though both arrived at the same answers).
Pascal used a recursive argument: the fair share at any stage equals the average of the fair shares after winning or losing the next round. This is the expected value approach.
Together, the two methods established the dual foundations of probability: the combinatorial/counting approach and the expectation/recursive approach.
Their correspondence also addressed the Chevalier de Mere's dice problem: how many throws of two dice to have a better than even chance of rolling double-six? (Answer: 25, not 24 as de Mere had guessed.)
Fermat developed a method he called adequality (from the Latin adaequalitas, inspired by Diophantus' parisotes) for finding maxima, minima, and tangent lines to curves.
To find the maximum of f(x):
f(x) ≈ f(x + e) (approximately equal)ee = 0This procedure is essentially computing f'(x) = 0, decades before Newton or Leibniz formalized the derivative. Fermat also extended it to find tangent lines, effectively computing derivatives of polynomials.
1. Set bx − x² ≈ b(x+e) − (x+e)²
2. Expand: bx − x² ≈ bx + be − x² − 2xe − e²
3. Cancel: 0 ≈ be − 2xe − e²
4. Divide by e: 0 ≈ b − 2x − e
5. Set e = 0: b − 2x = 0, so x = b/2
This is exactly what we get from setting the derivative equal to zero: f'(x) = b − 2x = 0.
Fermat also computed areas under curves y = x^n for integer and fractional n, anticipating the power rule of integration.
Intuitive guess from examples
Verify with numerical cases
(Or claim proof exists)
Send to correspondents
Fermat's position as a non-professional gave him freedom to pursue problems for their intrinsic beauty rather than institutional expectations. His focus on number theory — considered useless by contemporaries — would prove foundational centuries later.
Fermat's famous habit of writing results in book margins, with claims of proofs "too long for the margin," created a unique legacy. His son's posthumous publication of these marginalia turned them into challenges that drove number theory for 350 years.
Fermat proposed that light travels along the path requiring the least time. This elegant variational principle correctly derives Snell's law of refraction and became the prototype for all variational principles in physics (Hamilton, Lagrange, Feynman).
Fermat delighted in posing problems to other mathematicians, often knowing they were extremely difficult. His challenges to Wallis, Frenicle, and others stimulated research and created a competitive mathematical culture.
Fermat's most famous marginal note, written around 1637 in his copy of Diophantus' Arithmetica, stated that xⁿ + yⁿ = zⁿ has no positive integer solutions for n ≥ 3, and that he had found "a truly marvelous proof" that the margin was too narrow to contain.
For 358 years, the world's best mathematicians attempted to prove this. The theorem resisted all attacks until Andrew Wiles, building on the Taniyama-Shimura conjecture and the work of Ken Ribet, Gerhard Frey, and others, finally proved it in 1995.
Did Fermat actually have a proof? Almost certainly not for the general case. He likely had a proof for n = 4 (by infinite descent) and may have mistakenly believed his method generalized.
Euler proved n=3 (1770), Dirichlet and Legendre proved n=5 (1825), Lame proved n=7 (1839), Kummer proved it for all "regular" primes (1850s). Each case required new mathematics.
Wiles' proof used modular forms, elliptic curves, and Galois representations — mathematics entirely unknown to Fermat. The proof runs over 100 pages and required six years of secret work.
The quest to prove the Last Theorem drove the development of algebraic number theory, ideal theory (Kummer), class field theory, and the Langlands program — some of the deepest mathematics ever created.
Fermat's problems drove Kummer to invent ideal numbers, leading to Dedekind's ideals and the entire modern framework of algebraic number theory and commutative algebra.
Fermat's Little Theorem is the mathematical foundation of RSA encryption, the most widely used public-key cryptosystem. Every secure internet transaction relies on Fermat's 1640 discovery.
Fermat's principle of least time became the prototype for variational methods: Hamilton's principle, Lagrangian mechanics, and Feynman's path integral formulation of quantum mechanics.
The Pascal-Fermat correspondence established the mathematical framework that Bernoulli, Laplace, and Kolmogorov would develop into the modern theory of probability and statistics.
The quest to prove Fermat's Last Theorem contributed to the Langlands program — a vast web of conjectures connecting number theory, geometry, and representation theory, sometimes called "a grand unified theory of mathematics."
Wiles' proof connected Fermat's problem to elliptic curves and modular forms. Elliptic curve cryptography is now a leading method for digital signatures and key exchange.
Every time you visit an HTTPS website, Fermat's Little Theorem is at work. RSA encryption relies on modular exponentiation, where Fermat's theorem ensures that decryption reverses encryption. Billions of transactions per day depend on this.
Fermat's principle of least time explains reflection, refraction, and the design of lenses and optical instruments. It remains the foundation of geometric optics and is used in designing everything from cameras to fiber optic cables.
Fermat's Little Theorem provides a fast probabilistic primality test. Though not foolproof (Carmichael numbers are false positives), it forms the basis of the Miller-Rabin test used in generating cryptographic keys.
Modular arithmetic based on Fermat primes appears in hash functions, checksums, and error-detection codes. The Fermat primes (3, 5, 17, 257, 65537) are used in RSA public exponents.
Simon Singh (1997). Accessible account of the 350-year quest to prove Fermat's Last Theorem, culminating in Wiles' triumph. Excellent narrative of mathematical obsession.
Harold M. Edwards (1977). Rigorous mathematical history tracing the theorem through Euler, Kummer, and modern approaches. For readers with some mathematical background.
Michael Sean Mahoney (1994). The definitive scholarly biography, covering all aspects of Fermat's mathematical work in detail, including his methods and correspondence.
Paul Hoffman (1998). Biography of Paul Erdos that illuminates the number-theoretic tradition Fermat founded, showing how his problems continue to inspire mathematicians.
G.H. Hardy & E.M. Wright. Classic textbook that treats many of Fermat's theorems rigorously, placing them in the context of modern number theory.
"I have found a truly wonderful proof of this, which this margin is too narrow to contain."
— Pierre de Fermat, marginal note in Bachet's Diophantus (c. 1637)Pierre de Fermat (1601–1665)
Magistrate • Number Theorist • Prince of Amateurs