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 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 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 integers that can occur as the largest denominator in a representation of one by distinct unit fractions have density one.
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.
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.
Asks whether every finite coloring of the integers has distinct same-colored a, b, c with the reciprocal of a equal to the reciprocal of b plus that of c.
Bounds the fewest distinct unit fractions needed to represent any fraction with denominator b, and asks whether it is at most a constant times log log b; answered yes by the OpenAI release's Theorem 1.1 (2026), accepted on Lean.
Bounds the least possible largest denominator needed to write any fraction with denominator b by distinct unit fractions, against b times a power of log b.
The smallest integer not a sum of distinct unit fractions with denominators up to N, and whether the representable integers form an initial segment of integers; corrected to ask, for large N, whether that integer is the floor of the harmonic sum or one more.
Counts how many integers are sums of distinct unit fractions with denominators up to N, and asks whether there are only o(log N) of them.
Asks whether every subset of one through N of density at least alpha has a subset whose reciprocals sum to a rational with boundedly small denominator.
Asks how small the excess above one can be for reciprocals of consecutive integers from n summed until reaching one, and if n squared times it nears zero.
Asks whether every other increasing sequence whose reciprocals sum to one has liminf of its nth term raised to the power one over two to the n below 1.264085.
Asks whether a finite set of integers above one whose reciprocals sum to less than two can always be split into two parts each with reciprocal sum below one.
Asks whether every non-constant assignment of plus and minus one on an arithmetic progression has a finite subset whose signed reciprocals sum to zero.
Estimates how many distinct values arise as sums of reciprocals of subsets of the integers one through N.
The largest subset of the first N integers all of whose subsets have distinct sums of reciprocals.
Asks whether a set in which each n is a sum of two distinct elements at most C times splits into boundedly many parts with fewer than C each; the site's count of ordered pairs makes C = 2 trivial. Nešetřil and Rödl answer no.
Asks whether a minimal basis of positive density exists in which, for each of its elements, the integers needing that element have positive upper density.
Asks whether two sets of integers, each with counting function at least a constant times the square root of N, must share infinitely many equal nonzero differences.