Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
1,221 problems
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether a good sequence of pairwise coprime integers with convergent reciprocal sum can grow only polynomially, or at most subexponentially.
Asks how fast a sequence must increase if, for every n, only finitely many members a make n+a squarefree (property P), or if for infinitely many n every member a<n makes n+a squarefree (property Q).
Determines how fast an infinite set of integers must grow if every sum of two of its members is squarefree.
Estimates the largest chromatic number possible for a triangle-free graph on n vertices.
Asks for the anti-Ramsey numbers of cycles and paths, the most colors on the edges of the complete graph on n vertices without a rainbow copy: an asymptotic formula for cycles and an exact formula for paths.
Asks whether the number of distinct prime factors of the product of the partition numbers up to n tends to infinity, and eventually exceeds n.
Asks whether every large integer is the sum of at most r plus one numbers divisible by the r-th power of each of their prime factors, for r at least 2.
Asks whether the set of sums of distinct factorials contains only finitely many k-th powers for k at least 2, and only finitely many powerful numbers.
Estimates the largest subset of the numbers up to N all of whose pairwise sums are squarefree, and whether its size stays below every fixed power of N.
Concerns which integers are sums of numbers of the form a power of p times a power of q, none dividing another, for coprime integers p greater than q.
Asks whether bounded clique number and large chromatic number force two anticomplete vertex sets both of large chromatic number; the El-Zahar-Erdős problem, open beyond the case of chromatic number three.
Asks whether, for given gap bounds and k at least 3, every sufficiently lacunary sequence misses the k-fold sumset of some sequence with those gaps.
Concerns Sierpinski numbers, odd m for which two to the k times m plus one is never prime, and the sets of primes that divide all those values.
Concerns real polynomials of degree n whose roots are all real and form an arithmetic progression.
The linear-length asymptotic-path conjecture fails at every finite order, even for functions growing arbitrarily close to the logarithmic-square threshold that guarantees radial paths.
Asks whether some meromorphic or entire function has, for every two distinct values, the ratio of their solution counts in growing discs unbounded.
Concerns the number of points on the circle of radius r at which a non-monomial entire function attains its maximum modulus.
Concerns non-constant entire functions for which the set where the modulus exceeds some constant has finite measure.
Asks whether a family of entire functions taking at most m distinct values at each point has cardinality at most m, for m between countable and continuum.
Asks for the shortest path from zero to the unit circle inside the set where a monic polynomial with all roots in the unit disc has modulus at most one.
Asks whether circles in the plane that no disjoint line separates can always be covered by a single circle whose radius is the sum of their radii.
Asks whether an additive function that decreases from n to n plus one only for a density-zero set of n must be a constant multiple of the logarithm.
Compares the Boolean algebra of sets of integers modulo density zero with the Boolean algebra of sets modulo logarithmic density zero.
Asks whether a square and a circle of the same area can be cut into finitely many congruent pieces.
Asks whether a real function with twice its value at a point at most the sum of its values at two later equally spaced points must be monotonic.