(mathematics) The proven conjecture that a regular hexagonal grid, or honeycomb, is the best way to divide a surface into regions of equal area with the smallest total perimeter.

The theorem that the only simply connected, closed 3-dimensional manifold is a sphere.

(mathematics) The conjecture that the process of taking a number n and, if n is even, dividing it by 2, or if n is odd, multiplying it by 3 and adding 1, then repeating this process, will eventually yield 1.

(number theory) Given coprime positive integers a, b and c, such that a + b = c, and d the radical of abc (the product of its distinct prime factors), the conjecture that d is usually not much smaller than c (in other words, that if a and b are divisible by large powers of primes, then c usually is not).

(mathematics) A conjecture stating that, if Aˣ+Bʸ=Cᶻ, where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor.

(mathematics) A conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The first term in every such sequence appears to be 1.

(mathematics, physics) An attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics.

(algebra) The conjecture that any endomorphism of a Weyl algebra is an automorphism.

(mathematics) A famous problem on polynomials in several variables.

(number theory) A conjecture stating that every even integer greater than 2 can be expressed as the sum of two primes, which has been shown to hold up through 4 × 10¹⁸, but remains unproven.