Collatz conjecture
Choose a positive whole number. Halve it if it is even; otherwise triple it and add one. Repeat. Does every starting number reach one?
Number theory
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Riemann hypothesis
There are 25 primes up to 100 and 168 up to 1,000. Their uneven counts follow a broad trend; a conjecture about the zeta function predicts how far they can stray from a refined estimate.
Birch and Swinnerton-Dyer conjecture
An elliptic curve has both rational solutions and an analytic function. The conjecture predicts that these two ways of studying it encode the same rank.
Goldbach conjecture
Every even whole number greater than two appears to be the sum of two primes. A proof covering all of them is still missing.
Twin prime conjecture
Prime pairs such as 11 and 13 are separated by two. Do such pairs keep appearing without end?
Odd perfect numbers
A perfect number equals the sum of its positive proper divisors. All known examples are even; the existence of an odd one remains unsettled.
Lehmer’s Mahler measure problem
A polynomial’s roots define a measure of its size. Can integer polynomials have this measure arbitrarily close to one from above?
Littlewood conjecture
Two real numbers can each be approximated by fractions. The conjecture predicts especially good simultaneous approximations using a common denominator.
Schanuel’s conjecture
The conjecture predicts how many algebraically independent numbers must appear when independent complex numbers are paired with their exponentials.
Bombieri–Lang conjecture
Algebraic spaces of general type are predicted to have rational points confined to a smaller algebraic subset, rather than spread densely everywhere.
Hilbert’s tenth problem over the rationals
Can an algorithm always decide whether an equation with integer coefficients has a solution made of rational numbers?
Erdős–Straus conjecture
Can every fraction 4/n, with n at least 2, be written as the sum of three reciprocals of positive integers?
Legendre’s conjecture
Must a prime number occur between n squared and the next integer’s square, however large n becomes?
Artin’s primitive root conjecture
For a fixed eligible integer, how often do its powers generate every nonzero residue modulo a prime?
Leopoldt’s conjecture
Do the independent units of a number field remain independent when measured using p-adic arithmetic?
Sarnak’s conjecture
Does the arithmetic Möbius sequence average to zero against every sequence generated by a system with zero topological entropy?
Euler–Mascheroni constant
The difference between the harmonic series and a logarithm approaches a famous constant. Is that constant a ratio of integers?
Erdős–Moser equation
Can a power of an integer equal the sum of the same powers of every smaller positive integer, apart from 1 + 2 = 3?
Mersenne primes
Do prime numbers of the form 2 to a prime exponent, minus 1, continue forever?
Brocard–Ramanujan problem
Can a factorial plus one be a perfect square beyond the three known examples?
Singmaster’s conjecture
Is there a fixed limit on how often any number greater than one can appear in Pascal’s triangle?
Erdős–Ulam problem
Can points be dense throughout the plane while every distance between two of them is rational?
Deterministic prime construction
Can an algorithm reliably construct a prime of any requested bit length in polynomial time without randomness?
Inverse Galois problem
Can every finite group occur as the Galois group of a finite extension of the rational numbers?