Arithmetic geometry
Arithmetic questions studied through geometric objects.
10 problemsNumber theory
Integers, prime numbers, and arithmetic structures.
Explore Number theory
Collections
Browse collectionsPrimes and integers
Patterns in prime numbers, and equations whose solutions must be whole numbers.
14 problemsPrime patterns
Prime pairs, prime sums, and the distribution of primes through the lens of the zeta function.
7 problemsDiophantine equations
Equations over integers and rational numbers, from elliptic curves to the limits of decision algorithms.
7 problemsA few problems to explore
36 totalCollatz 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?
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.