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Fermat's theorem

Several theorems and principles in mathematics are named after the French mathematician Pierre de Fermat (1607–1665), who made foundational contributions to number theory, calculus, probability, and optics. "Fermat's theorem" is an ambiguous term that can refer to any of these results, with the most famous being Fermat's Last Theorem. This article provides an overview of Fermat's key ideas and discusses the following: In number theory: In calculus:

Interior Extremum Theorem

In optics:

Fermat's Principle

Detailed historical context and proofs are covered in subsequent sections.

Historical Context

Pierre de Fermat's Contributions

Pierre de Fermat was born in 1607 in Beaumont-de-Lomagne, France, and died on January 12, 1665, in Castres.^[1]^[2] He pursued a career as a lawyer and government official, qualifying in 1631 and serving as a councillor at the Parlement of Toulouse, where he rose through the ranks with promotions in 1638 and 1652, while pursuing mathematics as an amateur passion alongside his professional duties.^[2] Despite his primary occupation in law, Fermat's mathematical pursuits were profound and influential, often conducted through private correspondence rather than formal publications.^[2] Fermat engaged in significant exchanges with leading intellectuals of his time, including Blaise Pascal, René Descartes, and Marin Mersenne, which shaped early developments in several fields. His 1654 letters with Pascal laid the foundations of probability theory, addressing problems like the division of stakes in interrupted games.^[2] With Descartes, he debated methods in optics and geometry, contributing independently to analytic geometry through techniques for finding tangents and extrema of curves.^[2] Correspondence with Mersenne, starting in 1636, covered topics such as spirals and optimization problems, fostering the dissemination of his ideas within Europe's mathematical community.^[2] A hallmark of Fermat's approach was his method of infinite descent, a proof technique by contradiction that assumes a minimal counterexample and derives a smaller one, leading to an infinite regress.^[2] He famously annotated books with theorems stated sans proofs, most notably claiming in the margin of Diophantus's Arithmetica to have a proof too large for the space, sparking centuries of mathematical inquiry.^[2] Much of his work remained unpublished during his lifetime, shared instead via letters, which delayed but ultimately amplified his impact. Fermat's unpublished notes and correspondences profoundly influenced number theory, with innovations like results on primes and quadratic forms; precursors to calculus through his adeptness in maxima, minima, and tangents; and optics via principles governing light paths.^[2] His ideas, such as those in number theory exemplified by what became known as Fermat's Little Theorem, inspired subsequent generations and remain foundational in modern mathematics.^[2]

Evolution of Fermat's Ideas

Following Fermat's death in 1665, his son Samuel de Fermat compiled and published a new edition of Claude-Gaspard Bachet's 1621 Latin translation of Diophantus's Arithmetica in 1670, incorporating his father's extensive marginal annotations that revealed many of Fermat's unpublished number-theoretic insights.^[2] This posthumous volume disseminated Fermat's ideas widely, including his famous marginal note on the "last theorem," which stemmed from his habit of asserting results in correspondence or notes without providing proofs.^[2] Gottfried Wilhelm Leibniz engaged with Fermat's number theory in the late 17th century, developing an unpublished proof of Fermat's Little Theorem using the binomial theorem around 1676–1683, which helped formalize the result for later mathematicians.^[3] Leonhard Euler then advanced this further by publishing the first rigorous proof of the theorem in 1736 in his paper "Theorematum quorundam ad numeros primos spectantium demonstratio I," employing group-theoretic ideas avant la lettre to establish the result for prime moduli.^[4] Euler also contributed partial progress on Fermat's Last Theorem, proving it for exponent 3 via infinite descent in a 1753 letter to Christian Goldbach and including the argument in his 1770 textbook Vollständige Anleitung zur Algebra, though the proof contained a subtle gap later corrected by others.^[5] In the 19th century, Peter Gustav Lejeune Dirichlet mentored Ernst Eduard Kummer, encouraging his work on algebraic number theory; in 1847, Dirichlet reviewed Kummer's initial flawed attempt at a general proof of Fermat's Last Theorem, prompting Kummer to develop the concept of ideal numbers in 1844–1847 to restore unique factorization in rings of cyclotomic integers.^[6] Kummer's ideal numbers enabled proofs of the Last Theorem for all regular prime exponents, marking a pivotal advancement in Fermat's number theory by bridging Diophantine equations with abstract algebra.^[7] The Paris Académie des Sciences announced a competition in 1850 offering 3,000 francs for a general proof of Fermat's Last Theorem, with Joseph Liouville serving on the judging committee alongside Augustin-Louis Cauchy; no complete solution emerged, but Liouville's earlier 1847 critique of Gabriel Lamé's flawed proof attempt highlighted factorization issues in complex domains, influencing the field's direction despite the prize remaining unawarded for a full resolution.^[8]

Number Theory Theorems

Fermat's Little Theorem

Fermat's Little Theorem states that if p is a prime number and a is an integer not divisible by p, then a^{p-1} \equiv 1 \pmod{p}.^[9] An equivalent form is a^p \equiv a \pmod{p}, which holds even if p divides a.^[9] This congruence captures a fundamental property of primes in modular arithmetic, highlighting the multiplicative order of a modulo p dividing p-1.^[9] Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to Bernard Frénicle de Bessy, claiming a proof without providing details.^[10] No record of Fermat's proof survives, and the first published proof appeared nearly a century later from Leonhard Euler in 1736, using an inductive argument.^[4] Euler later provided additional proofs, including one in 1750 via the binomial theorem applied to (a + b)^p \equiv a^p + b^p \pmod{p} for prime p, and another in 1761 using precursors to group theory by considering cosets of subgroups in the multiplicative group modulo p.^[9] These proofs laid groundwork for modern abstract algebra, with the group-theoretic approach aligning with Lagrange's theorem on subgroup orders dividing the group order.^[9] The theorem underpins key applications in computational number theory. It forms the basis of the Fermat primality test, where for an odd integer n > 1, selecting bases a coprime to n and checking if a^{n-1} \equiv 1 \pmod{n} can prove compositeness if the congruence fails, as it must hold for primes.^[11] Composites passing the test for certain a are called Fermat pseudoprimes to base a, such as 341 to base 2, and absolute pseudoprimes (Carmichael numbers) like 561 pass for all coprime a, though they are rare and the test remains probabilistically effective.^[11] Furthermore, Euler generalized the theorem in 1763 to Euler's theorem: if \gcd(a, n) = 1, then a^{\phi(n)} \equiv 1 \pmod{n}, where \phi is Euler's totient function, reducing to Fermat's Little Theorem when n = p is prime since \phi(p) = p-1.^[12]

Fermat's Last Theorem

Fermat's Last Theorem asserts that there are no positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for any integer n > 2.^[13] This Diophantine equation generalizes the Pythagorean theorem, which holds for n=2, but prohibits solutions in positive integers for higher exponents.^[5] The theorem originated from a marginal note by Pierre de Fermat in his copy of Diophantus's Arithmetica around 1637, where he claimed to have discovered a "truly remarkable proof" that the margin was too narrow to contain.^[5] Fermat's son published the note after his death in 1665, sparking centuries of mathematical inquiry despite no surviving proof from Fermat himself.^[14] Early progress focused on specific exponents: Leonhard Euler provided a proof for n=3 in 1753, though it contained a subtle gap later addressed, using infinite descent in the ring of Eisenstein integers.^[5] In 1825, Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre independently completed the proof for n=5, handling various cases through factorization in cyclotomic fields.^[5] In the 19th century, Ernst Kummer advanced the general case by developing the theory of ideal numbers to restore unique factorization in cyclotomic integers, proving the theorem for all regular primes (those not dividing the class number of the (p-1)-th cyclotomic field) by 1850.^[15] This covered infinitely many exponents but left irregular primes unresolved, motivating deeper studies in algebraic number theory.^[16] The 20th century saw stalled attempts, such as those involving class field theory, until Gerhard Frey's 1986 insight linked hypothetical solutions to semistable elliptic curves of an anomalous form, suggesting they would contradict the Taniyama-Shimura conjecture if non-modular.^[17] Andrew Wiles announced a proof in 1993 by establishing the modularity theorem for semistable elliptic curves over the rationals, implying Frey's curves must be modular and thus nonexistent, thereby proving the theorem.^[18] An error in the Euler system was identified in 1994, which Wiles and Richard Taylor resolved using a 3-5 switch in the Galois representations, leading to the final publication in 1995.^[17] The proof spans over 100 pages and integrates elliptic curves, modular forms, and deformation theory, confirming no solutions exist for n > 2.^[18] The theorem's pursuit has profoundly influenced mathematics, driving the creation of algebraic number theory through Kummer's ideals and inspiring prize competitions like the 1908 Wolfskehl Prize of 100,000 German marks (equivalent to about $50,000 when claimed by Wiles in 1997).^[8] Its resolution not only settled a 358-year-old conjecture but also catalyzed advancements in the Langlands program and modern arithmetic geometry.^[19]

Fermat's Theorem on Sums of Two Squares

Fermat's theorem on sums of two squares states that an odd prime p can be expressed as p = x^2 + y^2 for some integers x and y if and only if p \equiv 1 \pmod{4}. The even prime 2 is also a sum of two squares, specifically $2 = 1^2 + 1^2. This characterization provides a precise condition for when primes admit such a representation, distinguishing them from primes congruent to 3 modulo 4, which cannot be written in this form.^[20] Pierre de Fermat first stated the theorem in a letter to Marin Mersenne on December 25, 1640, claiming a proof via his method of infinite descent but providing no details. The claim remained unproven until Leonhard Euler established it rigorously in 1749, with the proof appearing in print in his 1773 paper "Demonstratio theorematis Fermatiani de numeris primis formae $4n + 1 in summam duorum quadratorum resolubilibus." Euler's approach relies on infinite descent, showing that if a prime p \equiv 1 \pmod{4} divides a sum of two squares, then it must itself be such a sum, leading to a contradiction unless the representation is primitive.^[21] A sketch of Euler's proof begins by assuming p \equiv 1 \pmod{4} divides some m^2 + n^2 where m and n are integers not both divisible by p. Using properties of sums of squares and modular arithmetic, one constructs smaller positive integers a, b such that p divides a^2 + b^2, with a^2 + b^2 < p. Repeating this process yields an infinite descending sequence of positive integers bounded below by 1, which is impossible, implying p itself must be a sum of two squares. The "if" direction follows from the fact that squares modulo 4 are 0 or 1, so their sum cannot be 3 modulo 4.^[22]^[23] In modern terms, the theorem finds a natural interpretation in the ring of Gaussian integers \mathbb{Z} = \{ a + bi \mid a, b \in \mathbb{Z} \}, where the norm N(\alpha) = a^2 + b^2 is multiplicative. Primes p \equiv 3 \pmod{4} remain prime in \mathbb{Z}, while those with p \equiv 1 \pmod{4} factor as p = (a + bi)(a - bi) with norm p, ensuring unique factorization up to units implies such a representation. This perspective, developed later by Dedekind, underscores the theorem's role in algebraic number theory.^[24]^[25] The theorem extends to all positive integers: a number n can be written as a sum of two squares if and only if every prime congruent to 3 modulo 4 in its prime factorization has even exponent, with the prime 2 and primes congruent to 1 modulo 4 appearing to any power. This follows from the multiplicativity of the norm in Gaussian integers and the identity (x^2 + y^2)(u^2 + v^2) = (xu - yv)^2 + (xv + yu)^2, allowing products of such representations. The result has applications to quadratic reciprocity, as the condition p \equiv 1 \pmod{4} is equivalent to -1 being a quadratic residue modulo p, a key case in Gauss's law.^[26]^[27]

Fermat's Right Triangle Theorem

Fermat's right triangle theorem asserts that no right-angled triangle with rational side lengths can have an area that is a perfect square of a rational number.^[28] This non-existence result was proved by Pierre de Fermat using the method of infinite descent and was published posthumously in 1670 as part of his collected works.^[29] The theorem implies, in particular, that 1 is not a congruent number, meaning there exists no right triangle with rational sides and area equal to 1.^[28] Fermat's proof proceeds by assuming the existence of such a triangle with integer sides (after clearing denominators) and integer square area, then deriving a contradiction via infinite descent. Specifically, the proof leverages the parametrization of primitive Pythagorean triples, which generate the integer sides a = m^2 - n^2, b = 2mn, c = m^2 + n^2 for coprime integers m > n > 0 of opposite parity. The area \frac{1}{2}ab = mn(m^2 - n^2) is assumed to be a perfect square k^2. By analyzing the prime factors and applying descent, Fermat shows that this assumption leads to a smaller positive integer solution of the same form, resulting in an infinite descending sequence of positive integers, which is impossible.^[29]^[28] This descent on Pythagorean triples or related cubic forms establishes the theorem rigorously within the framework of 17th-century number theory.^[29] In modern terms, the theorem is intimately connected to the arithmetic of elliptic curves. The existence of a rational right triangle with area 1 is equivalent to the existence of a non-torsion rational point on the elliptic curve y^2 = x^3 - x, defined over the rationals.^[29] This Mordell curve has conductor 32 and complex multiplication by i; its group of rational points is precisely the torsion subgroup \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, consisting of the three points (-1, 0), (0, 0), and (1, 0).^[29] By the Mordell-Weil theorem, the full rational points form a finitely generated abelian group, and here the rank is 0, confirming no infinite-order points and thus no such triangle.^[29] Fermat's descent anticipates the 2-descent procedure on elliptic curves, providing an elementary precursor to these advanced tools.^[28] The infinite descent technique in Fermat's proof shares methodological similarities with the approach used to establish the case n=3 of Fermat's Last Theorem, where no non-trivial positive integer solutions exist to x^3 + y^3 = z^3.

Calculus Theorems

Interior Extremum Theorem

The Interior Extremum Theorem, also known as Fermat's theorem on stationary points, asserts that if a function f has a local maximum or minimum at an interior point c in its domain and f is differentiable at c, then the derivative f'(c) = 0. This condition implies that any local extremum of a differentiable function occurs at a critical point where the tangent is horizontal. Pierre de Fermat introduced this result in his circa 1636-1637 manuscript Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum, where he developed a method for locating maxima and minima of algebraic functions using the concept of "adequality" (adaequalitas), a precursor to taking limits.^[2] Fermat's approach involved substituting a small increment e into the function, setting up an approximate equality f(a + e) \approx f(a), canceling common terms, dividing by e, and neglecting higher-order terms in e to derive a condition equivalent to the zero derivative at the extremum. This work, communicated through letters to Marin Mersenne and other mathematicians, predated the formal invention of calculus by Isaac Newton and Gottfried Wilhelm Leibniz by several decades and drew criticism from René Descartes, positioning Fermat as a key figure in the early development of infinitesimal methods.^[30] In its historical context, Fermat's approach emerged as part of broader efforts in the 17th century to find tangents to curves and solve optimization problems geometrically, building on algebraic techniques from ancient sources like Diophantus while innovating with near-equality arguments.^[31] The theorem remained in manuscript form until its publication in Fermat's collected works in the 19th century, where it was rigorously formalized within the framework of modern calculus, influencing texts on differential analysis.^[2] The theorem serves as the foundational principle for first-derivative tests in optimization, enabling the identification of potential local extrema by solving f'(x) = 0, and underpins applications in analytical optimization problems across mathematics and physics.

Optics Principles

Fermat's Principle

Fermat's principle in optics states that the path taken by a light ray between two given points is the one that can be traversed in the least time, or more generally, the path for which the time is stationary (a minimum, maximum, or saddle point). This principle is equivalent to light following the path of stationary optical path length, defined as the integral of the refractive index along the path.^[32] Pierre de Fermat first formulated this principle in 1657 in a letter to Marin Cureau de La Chambre, proposing that the law of refraction could be derived from a minimum-time assumption, contrasting with Descartes' earlier instantaneous propagation model. Fermat's idea built on Hero of Alexandria's least-distance principle for reflection but extended it to time minimization, assuming finite light speed in different media. This formulation anticipated variational methods, deriving Snell's law by optimizing the time for paths crossing a refractive interface.^[33] The principle is rigorously derived using the calculus of variations, where the travel time t for a ray from point A to B in a medium with refractive index n is given by

t = \frac{1}{c} \int_A^B n \, ds,

with c the speed of light in vacuum and ds the infinitesimal path element. Stationary paths satisfy the Euler-Lagrange equation from minimizing this functional, leading to Snell's law n_1 \sin \theta_1 = n_2 \sin \theta_2 at interfaces. For continuous media, the rays follow the eikonal equation |\nabla S| = n, where S is the optical path.^[32]^[34] In wave optics, Fermat's principle emerges as the stationary-phase approximation to Huygens' principle, where ray paths correspond to directions of constructive interference from secondary wavelets, ensuring the phase is extremal along the path. This connection unifies ray and wave descriptions, with Fermat's paths tracing the envelope of Huygens' wavelets in the geometric limit. Modern applications include fiber optics, where meridional and skew rays in graded-index fibers follow stationary-time trajectories to minimize dispersion and signal attenuation, and general relativity, where null geodesics in curved spacetime obey a Fermat-like principle for light propagation in gravitational fields.^[35]^[36]^[37]

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