Open problems by topic
Topics group problems across fields and sources by what they are about. Each topic page lists every open problem with that subject, most active first.
Prime numbers
Questions about how primes are distributed: gaps between consecutive primes, primes in special forms and sequences, and sums and products involving primes.
36 problemsRamsey theory
Ramsey theory asks how large a structure must be before some order is unavoidable — a monochromatic clique in a coloured graph, a monochromatic arithmetic progression, a solution to an equation inside one colour class.
26 problemsAdditive combinatorics
Sumsets, arithmetic progressions in dense sets, sum-free sets and the structure of sets with few sums.
19 problemsDistance problems
How many distinct distances must n points determine, how often can one distance repeat, and related questions in the plane and higher dimensions.
17 problemsSidon sets
A Sidon set has all pairwise sums distinct.
15 problemsUnit fractions
Writing numbers as sums of distinct fractions 1/n: which denominators suffice, how many terms are needed, and what happens when the denominators are restricted.
12 problemsAdditive bases
A set is an additive basis if every large integer is a sum of a bounded number of its elements.
11 problemsDivisors
Problems on the divisors of integers: their distribution, consecutive divisors, divisor sums and highly composite numbers.
10 problemsChromatic number
How many colours a graph or hypergraph needs, and what forces that number up: girth, forbidden subgraphs, geometric constraints.
10 problemsArithmetic progressions
When must a set of integers contain an arithmetic progression, and how large can a set without one be? From van der Waerden numbers to density bounds, with small cases open to computer search..
10 problemsIrrationality
Is a given series or constant irrational, or transcendental? Most problems here are Erdős's questions about series built from integer sequences; partial results and numerical evidence are both useful..
10 problemsBinomial coefficients
Prime factors, divisibility and multiplicative structure of binomial coefficients.
10 problemsPowerful numbers
A number is powerful if every prime dividing it divides it at least twice.
9 problemsFactorials
Equations and divisibility questions involving n!, products of consecutive integers, and their prime factors..
8 problemsPermutation patterns
Counting permutations that avoid a pattern such as 1324, growth rates of permutation classes, and the enumeration sequences behind them.
7 problemsCovering systems
A covering system is a finite set of congruences that every integer satisfies.
7 problemsIterated functions
What happens when an arithmetic function such as φ, σ or a Collatz-type map is applied repeatedly? Orbits are easy to compute; proofs about all starting values are not..
13 problemsSet theory
Partition relations for infinite cardinals and ordinals, many of them from Erdős and his collaborators.
6 problemsHypergraphs
Turán-type, colouring and matching questions for hypergraphs, where far less is known than for graphs..
6 problemsCycles in graphs
Which cycle lengths must a graph contain given its edge count, minimum degree or chromatic number? Small counterexamples, if any exist, can be searched for exhaustively..