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.
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 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 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 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 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 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 nth prime divided by two to the n is irrational.
Estimates the largest lower bound for the maximum modulus on the unit circle of a product of terms one minus z to the a-i, over all choices of n exponents.
Asks whether the sum of one over two to the n minus one, taken over any infinite set of naturals, is irrational.
Asks for which n the value n over two to the n is a sum of distinct terms k over two to the k, and whether some rational has uncountably many such sums.
Asks whether two to the two to the n keeps reciprocal sums irrational under all asymptotically equal replacements, and whether such sequences must grow fast.
The powers-of-two case is false; asks whether factorial denominators keep their reciprocal sum irrational under every bounded nonzero integer perturbation.
Asks how fast an increasing integer sequence can grow when the sum of reciprocals of its terms and of its terms minus one are both rational.
Asks whether the sum of reciprocals of the least common multiples of the first n integers built from a fixed finite set of primes is irrational.
Determines explicitly, and bounds the growth of, the greedy sequence starting at 0 and n that avoids any three-term arithmetic progression.
The largest number of subsets of {1,...,N} with every pairwise intersection a non-empty arithmetic progression; open for the exact value, while Szabó's linear-error question, N^2/2 + O(N), has a Lean proof Conjectures.io accepted.
Asks whether there is a covering system of congruences whose moduli are all of the form p minus one for primes p at least 5.
Asks whether a group can be exactly covered by more than one coset when the cosets have different sizes, each element lying in exactly one.
Asks whether some Fibonacci-type sequence has all terms composite while no integer shares a factor with every term.
Asks whether the greedy algorithm picking the least allowed denominator always terminates for a rational with odd denominator when only odd ones are allowed.
Asks whether distinct denominators above one whose unit fractions sum to one must always include two consecutive ones differing by at least three.
Asks whether only finitely many pairs of integer intervals have their combined sum of reciprocals equal to a whole number.
Asks whether the harmonic sum numerator over the least common multiple of one through n is coprime to it infinitely often, and not coprime infinitely often.
Estimates the growth of the least integer above one that never appears as a denominator in any representation of one as a sum of k distinct unit fractions.