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 Fibonacci-type sequence has all terms composite while no integer shares a factor with every term.
Asks whether, for every c, some n has sum of divisors above c times n yet admits no covering system whose moduli are distinct divisors of n above one.
The maximum density of integers covered by choosing one congruence class for each modulus in a finite set, and whether equal classes minimize the density.
Asks whether, for every k at least 3, congruence classes can be chosen modulo each prime so that all large integers lie in one of them with quotient at least k.
Asks whether a sequence of moduli growing faster than k log k always leaves a number of uncovered integers below each modulus that is not small compared with k.
Asks whether moduli whose congruences always leave a density-zero set must have a finite initial segment leaving density below any given epsilon.
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, for a suitable polynomial, every large integer is the sum of its values over the denominators of some unit fraction representation of one.
Asks whether the largest possible smallest denominator among k distinct unit fractions summing to one is asymptotically k divided by e minus one.
Asks whether the least possible largest denominator among k distinct unit fractions summing to one is asymptotically e over e minus one, times k.
Asks whether k distinct unit fractions summing to one can always be found with all denominators inside an interval of width about e minus one times k.
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, for every large k, one can be written as the sum of reciprocals over k separated intervals of integers, each of length at least two.
Asks whether extending a block of consecutive reciprocals starting at a by one more term can lower its denominator in lowest terms, and how far one must go.
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.
Asks whether the integers that can occur as the largest denominator in a representation of one by distinct unit fractions have density one.
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.
Estimates the least starting value t for which one cannot be written as a sum of distinct unit fractions with denominators from t up to N.
Asks whether the fewest distinct unit fractions with denominators at least N summing to one exceeds e minus one times N by an amount tending to infinity.
Estimates the largest number of disjoint subsets of one through N whose reciprocals each sum to one, and asks whether it is o(log N), of smaller order than log N.
Counts the subsets of the integers one through N whose reciprocals sum to one.
Asks whether every set of positive integers of positive density contains a finite subset whose reciprocals sum to one.
Asks whether some infinite increasing sequence of integers with bounded gaps has no finite subset of reciprocals summing to one.
Estimates the size of the largest subset of one through N having no subset whose reciprocals sum to one.