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 least possible number of nonzero terms in the square of a rational polynomial with exactly k nonzero terms tends to infinity as k grows.
Asks whether every set of integers of positive density contains three distinct members one of which is the least common multiple of the other two; true by Kleitman's union-free theorem, attested here second-hand.
Asks whether two subsets of the first N integers with all pairwise products distinct must have size product at most about N squared over the logarithm of N; proved by Szemerédi (1976), with a second proof by Erdős and Szemerédi.
Asks whether an additive function whose consecutive differences stay bounded must equal a constant multiple of the logarithm plus a bounded error.
Asks whether, for a real sequence tending to infinity whose consecutive ratios tend to one, the positions of the multiples of almost every real within the sequence's gaps are uniformly distributed; LeVeque's question, disproved by Schmidt. The site's wording restricts to integer sequences, for which the Davenport–Erdős theorem gives yes.
Asks whether there is a fixed k such that every large enough integer is the product of k integers at least two minus their sum.
Asks whether, for k greater than two, the multiset of all sums of k distinct elements of a finite set of complex numbers determines the set, given its size.
Asks whether, for every positive irrational alpha, sums of two positive squares come arbitrarily close to alpha times a positive square; the site's wording over every irrational alpha fails at negative ones.
Determines the number of antichains of subsets of an n-element set, that is, families in which no member contains another.
Asks whether at most the middle binomial coefficient of the signed sums of n complex numbers of modulus at least one can lie in one open unit disc.
Asks whether every n by n doubly stochastic matrix has a permutation along which the product of entries is at least n to the power minus n.
Asks whether bounded sets of outer measure below one assigned to the reals leave an infinite set no member of which lies in another's set (independent of ZFC), and whether closed sets of measure below one leave three (proved).
Estimates the largest size of a set of points in n-dimensional space realizing only two distinct pairwise distances; settled to leading order: n^2/2 + O(n).
Determines the largest angle that is guaranteed to appear among some three points of every set of n points in the plane.
Asks whether every set of diameter one in n-dimensional space splits into at most n plus one pieces of smaller diameter.
Asks whether the set where a monic polynomial has modulus below one has boundedly many components of diameter above a fixed constant, whatever the degree.
Asks whether the mean absolute value of the exponential sum over a set of N integers is at least of order the logarithm of N.
Asks whether every entire nonpolynomial function has a rectifiable path to infinity along which the integral of any negative power of its modulus is finite.
Asks whether an entire function of finite order with very sparse exponents has minimum modulus whose logarithm matches that of its maximum modulus.
Asks whether the largest modulus among the first n power sums of complex numbers, one of which is one, is bounded below by an absolute positive constant.
Asks whether Rademacher random multiplicative sums over sqrt(N log log N) almost surely have a positive constant limit superior; a Lean proof built and audited here makes the ratio tend to 0, so the answer is no.
Asks whether, for a random polynomial of degree n with independent sign coefficients, the count of real roots divided by log n tends almost surely to two over pi.
Concerns the behavior of a random polynomial of degree n whose coefficients are chosen independently and uniformly from plus one and minus one.
Asks whether almost all degree n polynomials with coefficients plus or minus one dip below absolute value one on the unit circle, and how small that minimum is.
Characterizes which sequences of arc lengths tending to zero with infinite sum make random independent arcs cover the whole unit circle with probability one.