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

(geometry) A theorem stating that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.

(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.

(number theory) the theorem that states that every integer greater than one is uniquely expressible as a product of prime numbers, which is called its prime factorization

(geometry) A theorem which states that, in 3-dimensional space, any displacement of a rigid body such that some point on it remains fixed is equivalent to a single rotation about some axis that runs through said point.

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.

(number theory) A theorem stating that, if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.

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

(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.