(mathematics) A theorem about packing congruent rectangular bricks into larger rectangular boxes so that no space is left over. It states that a "harmonic brick" (one in which each side length is a multiple of the next smaller side length) can only be packed into a box whose dimensions are multiples of the brick's dimensions.

(geometry) A theorem stating that, given a finite number of points in the Euclidean plane, either all the points lie on a single line, or there is at least one line which contains exactly two of the points.

In dynamical systems theory, a theorem stating that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also known n first integrals of motion that are independent and in involution, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the canonical transform is explicitly known.

(mathematics) A particular refinement of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions.

(mathematics) A theorem showing that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups; one of the foundational results of anabelian geometry.

(mathematics) A theorem stating that for each positive integer d there is a finite set Y_d of prime numbers, such that if p is any prime not in Y_d then every homogeneous polynomial of degree d over the p-adic numbers in at least d²+1 variables has a nontrivial zero.

(mathematical analysis) A theorem which states that any real-valued Lebesgue integrable function can be approached arbitrarily closely from below by an upper semicontinuous function and also from above by a lower semicontinuous function.

(mathematical analysis) A theorem which states that for any subset S of an n-dimensional Euclidean space, S is compact if and only if it is both closed and bounded.

(mathematics) A theorem stating that a polynomial ring over a Noetherian ring is Noetherian.

(mathematics) A theorem stating that every finite group of odd order is solvable.