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 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.
Estimates the largest possible upper density of a measurable planar set containing no two points at distance one, and asks whether it is at most one quarter.
Asks whether the sum of the squares of the first N prime gaps is at most a constant times N times the square of the logarithm of N.
Asks whether the density of primes whose gap to the next prime is below c times the logarithm exists for every c and varies continuously in c.
Asks whether the distribution of normalized gaps between integers coprime to the product of the first k primes tends to a continuous limiting function.
Asks whether the number of ways to write n as a prime plus a power of two is always small compared with the logarithm of n.
Asks whether any set of integers with at least logarithmically many elements up to N gives some integers unboundedly many representations as prime plus member.
Asks whether for any positive constants there are, below every large x, more than a multiple of log x consecutive primes that are pairwise far apart.
Asks whether every multiplicative function taking only the values plus and minus one has a mean value.
Asks whether some infinite set of primes has the property that the gaps between consecutive integers built only from those primes tend to infinity.
Asks whether the largest subset of the first N integers whose three-element sums are all distinct has size asymptotic to the cube root of N.
Asks whether every integer greater than 2 has four over it written as a sum of three reciprocals of distinct positive integers.
Asks whether a sequence whose terms are asymptotically the square of the previous one, with rational reciprocal sum, must eventually satisfy a fixed recurrence.
Asks whether, for a constant greater than 1, the integers formed as a prime plus the integer part of a power of that constant have positive density.
Asks whether a sparse infinite set of naturals must have its sumset, counted up to N, at least three times as large as the set, up to o(1), along a sequence of N.
Asks whether, for coprime a and b, every large integer is a sum of distinct numbers of the form a to the k times b to the l.
Asks whether the sum of two to the power minus a-n is transcendental whenever the increasing integer sequence a-n has unbounded ratio to n.
Asks whether there are infinitely many n for which the number of distinct prime factors of n plus k stays of order k for every positive k.
Asks whether the sum over n of Euler's totient function of n divided by two to the n is irrational.
Asks whether the sum over n of the sum-of-divisors function of n divided by two to the n is irrational.