Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
606 of 1,221 problems match
Every problem posed by Paul Erdős, with its status, references, discussion and proof claims.
Asks whether the complement of a set of integers with all pairwise sums distinct must contain an infinite arithmetic progression.
Asks whether the complement of a set of reals with no three-term arithmetic progression must contain an infinite arithmetic progression.
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 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.
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 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 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.
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.
Asks whether some entire non-linear function sends exactly the rational real numbers to rational values.
Asks whether, for a non-polynomial entire function, the limiting ratio of largest coefficient term to maximum modulus, when it exists, must be zero.
Asks whether every large n admits a degree n polynomial with plus or minus one coefficients whose modulus stays within fixed multiples of the root of n on the unit circle; yes for every n at least 2 by Balister et al. (2020).
Asks whether, given sets of complex numbers with no finite limit point, some transcendental entire function has each set among zeros of some derivative.
Asks whether a polynomial with unimodular coefficients has maximum modulus on the unit circle exceeding the square root of its degree by a constant factor.
Asks whether every string of length 2^k over k letters contains two adjacent blocks that are permutations of each other; the site prints 2^k - 1, Erdős's misprint, and Keränen's word on four letters disproves it.