(mathematics) A theorem that establishes the transcendence of a large class of numbers, stating that, if a and b are algebraic numbers with a ≠ 0, 1, and b irrational, then any value of aᵇ is a transcendental number.

(mathematics) A result on the class group of certain number fields, strengthening Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity iff p divides the numerator of the n-th Bernoulli number Bₙ for some n, 0 < n < p − 1. The Herbrand–Ribet theorem specifies what, in particular, it means when p divides such an Bₙ.

(artificial intelligence, computing) A theorem that implies that no single machine learning algorithm is universally the best-performing algorithm for all problems.

(mathematics) A theorem stating that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.

(algebra) A theorem which states a constraint on rational solutions of a polynomial equation with integer coefficients.

(number theory) A theorem stating that the sequence of prime numbers contains arbitrarily long arithmetic progressions.

(mathematics) A fundamental combinatorial result of Ramsey theory, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure, and cannot be completely random.

(complex analysis) A theorem that gives a geometric relation between the roots of a polynomial P and the roots of its derivative P′. It states that the roots of P′ all lie within the convex hull of the roots of P.

(mathematics) A result on the combinatorics of block designs, stating that, if a (v, b, r, k, λ)-design exists with v = b (a symmetric block design), then: (i) if v is even, then k − λ is a square; (ii) if v is odd, then the following Diophantine equation has a nontrivial solution: x² − (k − λ)y² − (−1)^((v−1)/2) λ z² = 0.

Synonym of Kochen-Specker theorem