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.
Determines how large a subset free of k-term arithmetic progressions can be guaranteed inside any N integers, and how that compares with the case of one to N.
The largest number of congruence classes with distinct moduli at most N that can be chosen so that no integer lies in two of them.
Asks whether some integer coprime to 6 makes every number of the form two to the k times three to the l times it, plus one, composite.
Asks whether some integer has a covering system using its divisors above one whose classes overlap only for coprime pairs of moduli.
Asks whether every large integer is a power of two plus a number whose count of prime factors with multiplicity is below the iterated logarithm.
Asks whether, for almost every positive real number, the best sums of n distinct unit fractions below it are eventually built greedily.
Asks whether, for every g at least 2, large Steiner triple systems exist in which any j edges span at least j plus 3 vertices for j up to g.
Bounds the gaps between consecutive squarefree numbers, asking whether they are smaller than any fixed power, and whether a sharp logarithmic bound holds.
Asks whether at least four non-parallel lines with no four meeting at a point must form a triangle whose corners each lie on only two lines.
Asks whether the least number of lines through exactly two of n points, not all collinear, grows without bound, and how fast.
Asks whether n points in the plane with at most n minus k on any line always determine at least a constant times k times n lines through two or more points.
Asks whether the plane contains a dense set of points all of whose pairwise distances are rational.
Asks whether, for every n at least 4, there are n points in the plane with no three collinear and no four concyclic and all distances integers.
Asks whether the complement of any planar set that avoids distance one must contain the four corners of a unit square.
Asks whether some planar set has the property that every translated and rotated copy of it contains exactly one integer lattice point.
Asks whether enough points in general position in the plane always contain the vertices of an empty convex k-gon, and asks for an estimate of how many are needed.
Asks for which n there are n points, no three collinear and no four concyclic, whose distances take each multiplicity up to n minus one.
Asks whether consecutive prime gaps increase half the time and decrease half the time, and whether two consecutive gaps are equal infinitely often.
Asks whether the primes contain arithmetic progressions of every finite length.
Asks whether the sum of squared gaps between consecutive integers below n and coprime to n is at most a constant times n squared over Euler's totient of n.
Asks whether some set of integers with at most about N over log N elements up to N lets every large integer be a power of two plus one of its elements.
Bounds the gaps between consecutive integers that are sums of two squares, from above and below.
Estimates the largest number of pairs at distance one among n points of diameter one in d-dimensional space.
Asks whether any two to the d plus one points in d-dimensional space must include three that form an obtuse angle.
Asks whether a trigonometric polynomial with all real roots and maximum modulus one has integral of its absolute value at most 4 over a full period.