Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
516 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Estimates the largest number of pairs at distance exactly one among n points in d-dimensional space that are pairwise at distance at least one.
Estimates the largest possible number of pairs at distance exactly one among n points in d-dimensional space.
Estimates the largest number of triangles of equal area whose vertices come from a set of n points in the plane.
Estimates the largest number of four-point subsets with two pairs at equal distance among n points in the plane, and whether it is nearly n cubed.
Estimates how many points in d dimensions force n of them with all pairwise distances distinct, and whether that count is subexponential in d.
Studies the deficiency of n choose k, the number of the k integers from n downwards whose prime factors are all at most k, when no prime up to k divides it.
Asks whether the least prime factor of n choose k is at most the larger of n over k and k for all n at least 2k, with only finitely many exceptions.
Estimates the smallest n greater than k plus one for which every prime factor of n choose k exceeds k.
Determines how many values can arise as the common difference of a three-term arithmetic progression inside a set of n integers.
Concerns the number of consecutive pairs of divisors of n that are coprime.
Asks whether a good sequence of pairwise coprime integers with convergent reciprocal sum can grow only polynomially, or at most subexponentially.
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 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.
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.
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 for the least value, over n nodes in the interval from minus one to one, of the integral of the sum of squares of the Lagrange basis polynomials, and whether that least value is 2 minus (1+o(1))/n.
Asks for the best possible bounds on the fundamental Lagrange interpolation polynomials for nodes in the interval from minus one to one.
Asks whether every orbit of the shortcut Collatz map reaches one; open, with the site's caveat on the reported Erdős prize figure, verification below 2^71, no cycle with at most 91 local minima, and Tao's almost-all theorem.
Asks whether the largest product of two consecutive prime gaps below x is negligible compared with the square of the largest prime gap below x.