Mathematics · Curated list
Pythagorean theorem, Fermat's Last Theorem, Binomial theorem and 27 more, ranked by how much of the world has written about each one.
Most written-about first. Each description summarises the theorem's Wikipedia article, which its name links to.
1
Relation between sides of a right triangle
Also called Pythagoras's theorem, this is a basic relation of Euclidean geometry linking the three sides of a right triangle: the square built on the hypotenuse has an area equal to the combined areas of the squares built on the other two sides.
Wikidata · Photo: en:User:Wapcaplet, CC BY-SA 3.0
2
17th-century conjecture proved by Andrew Wiles in 1994
A result of number theory saying that no positive integers a, b, c and n, with n > 2, satisfy aⁿ + bⁿ = cⁿ. For n = 1 and n = 2, solutions without limit in number have been known since antiquity.
3
Algebraic expansion of powers of a binomial
A result of elementary algebra describing how powers of a binomial expand. It says (x + y)ⁿ becomes a polynomial made of terms axᵏyᵐ, in which k and m are nonnegative integers with k + m = n, and each coefficient a is a particular positive integer that depends on n and k.
4
Triangle area in terms of side lengths
A geometric formula that gives a triangle's area from the lengths of its three sides, a, b and c.
Wikidata · Photo: David Weisman ( Dweisman ), Public domain
5
Property of all triangles on a Euclidean plane
A trigonometric equation linking how long each side of any triangle is to the sine of each of its angles.
Integers have unique prime factorizations
Known too as the unique factorization theorem or the prime factorization theorem, it says that every integer above 1 is either a prime or can be written as a product of primes in exactly one way, apart from the order of the factors.
7
Generalization of Pythagorean theorem
A trigonometric rule that connects the side lengths of a triangle with the cosine of one of its angles.
8
Complex exponential in terms of sine and cosine
Named for Leonhard Euler, this formula of complex analysis sets out the basic tie between the complex exponential function and the trigonometric functions.
Wikidata · Photo: Original: Gunther Derivative work: Wereon, CC BY-SA 3.0
Mathematical rule for inverting probabilities
Named for Thomas Bayes, this rule of probability theory reverses conditional probabilities, so that the probability of a cause can be worked out from its effect. It lets you find, for instance, how likely a patient is to have a disease after a positive test, using how often the test comes back positive when the disease is there.
Relationship between derivatives and integrals
The theorem connecting differentiation of a function with integration of a function. Put loosely, it shows that each of the two operations undoes the other.
11
A prime p divides a^p–a for any integer a
A number-theory result: when p is prime, p divides aᵖ − a exactly for every integer a.
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12
Every polynomial has a real or complex root
Also known as d'Alembert's theorem or as the d'Alembert–Gauss theorem, it says that every non-constant polynomial in a single variable whose coefficients are complex numbers has a complex root, at least one. Real coefficients are no exception: any real number counts as a complex number with zero for its imaginary part.
13
Theorem in mathematics
A theorem of calculus and real analysis about differentiable functions. Roughly, over any interval there is a point where such a function's instantaneous rate of change matches its average rate of change across the interval: a car that travels a distance smoothly in a fixed time must, at some moment, be moving at exactly its average speed.
14
Planar maps require at most four colors
Also called the four-color map theorem, it says that four colours always suffice to colour every region of a map in such a way that neighbouring regions never share a colour, where neighbours are regions sharing a stretch of border of non-zero length.
Limitative results in mathematical logic
Two theorems of mathematical logic about the limits of what can be proved in formal axiomatic theories. Kurt Gödel published them in 1931, and they matter both to mathematical logic and to the philosophy of mathematics.
16
Mathematical rule for evaluating limits
A theorem for finding the limit when one function is divided by another and both tend to zero or to infinity: take the derivative of each first. It is named after Guillaume de l'Hôpital, a French mathematician of the 17th century, who printed it in a 1696 textbook; he had learned it from Johann Bernoulli, the Swiss mathematician who tutored him.
17
Pair of logical equivalences
Also known as De Morgan's theorem, they are a pair of transformation rules of propositional logic and Boolean algebra, both valid rules of inference. They take their name from Augustus De Morgan, a British mathematician of the 19th century.
18
Mathematical equation linking e, i and π
Wikidata · Photo: AstroOgier, CC BY-SA 4.0
19
Averages of repeated trials converge to the expected value
A law of probability theory: as more and more independent random samples are taken, the average of their results converges to the true value, where one exists. Formally, when values are independent and identically distributed, their sample mean tends to the true mean.
Wikidata · Photo: Dean P Foster, CC BY-SA 3.0
Theorem in combinatorics
The principle that if n items go into m containers and n > m, some container must hold more than one item. Among three gloves, for example, at least two are right-handed or at least two are left-handed, since three objects are shared between only two kinds of handedness.
Mathematical inequality relating inner products and norms
A bound from above on how large, in absolute value, the inner product of two vectors in an inner product space can be, given in terms of the product of their norms. It counts among the most important and most widely used inequalities in all of mathematics.
Theorem in real analysis
A theorem of calculus and real analysis named after Michel Rolle: a real-valued differentiable function that takes the same value at two different points must have a stationary point between them, a point where its derivative is zero.
24
Characterization of how many integers are prime
Often shortened to PNT, the theorem describes how prime numbers are distributed among the positive integers in the long run. It makes precise the intuition that primes thin out as numbers grow, by measuring exactly how fast that happens.
25
Counterintuitive result in probability
The theorem that a monkey striking typewriter keys at random, independently, for an infinite length of time will almost surely produce any given text, the complete works of William Shakespeare included. More precisely, if every keystroke is independent and random, every finite text that could exist would almost surely appear infinitely many times.
Wikidata · Photo: New York Zoological Society, Public domain
On triangles inscribed in a circle with a diameter as an edge
A result of geometry: take three distinct points A, B and C on a circle, with the line AC a diameter, and the angle ∠ ABC is always a right angle. The inscribed angle theorem contains it as a special case, and Euclid's Elements states and proves it as part of the 31st proposition of its third book.
Property of geometry, also used to generalize the notion of "distance" in metric spaces
The rule that in any triangle the lengths of two sides together are at least as great as the length of the third. Stated this way it allows degenerate triangles, though some authors, particularly on elementary geometry, rule them out and so drop the case of equality.
About simultaneous modular congruences
A theorem saying that if the remainders of an integer n divided by several integers are known, the remainder of n divided by their product is fixed uniquely, provided the divisors are pairwise coprime.
Formula for systems of linear equations
An explicit formula of linear algebra for solving a system of linear equations in which the number of equations equals the number of unknowns, whenever that system has exactly one solution. It gives the solution through determinants: that of the square coefficient matrix and those of matrices made from it by swapping one column for the column of right-hand sides.
Formula in calculus
A calculus formula for differentiating one differentiable function composed with another, z and y: it expresses that derivative using the derivatives of z and of y.
Every Wikidata item that is a theorem, or a kind of theorem, and that has its own English Wikipedia article, ranked by the number of Wikimedia sites with a page about it — Wikipedia's language editions, mostly, and sister projects such as Wikiquote. That counts how many communities independently thought it worth describing, and nobody can buy a place on it. The top 30 are shown.
Wikidata files these under this list's query, but they are not what the list is about:
Fetched from Wikidata on 2026-09-25. Each description is this site's summary of the entry's Wikipedia article, which its name links to: the wording is ours, the facts are Wikipedia's, licensed CC BY-SA 4.0. To correct an entry, correct it there; the next refresh carries the change.
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