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.
Estimates the largest subset of one through N in which no reciprocal is a sum of reciprocals of other distinct members, and asks whether it is about half of N.
Estimates the largest subset of one through N with no distinct members where one reciprocal is the sum of two others, and asks whether it is about half of N.
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.
Asks whether every positive rational with squarefree denominator is a sum of distinct unit fractions whose denominators are all products of two distinct primes.
Asks whether two finite sets of primes exist whose sums of reciprocals multiply together to give one.
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 whether the least non-zero distance from one to a subset sum of reciprocals of one through N decays like e to the power of minus a constant times N.
Asks whether a large multiset of integers whose reciprocals sum above K always has a subset whose reciprocals sum to at most one but within e to the minus cK.
Asks whether infinitely many sums of reciprocals of distinct primes equal one minus the reciprocal of an integer.
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 signs of minus one, zero or one can always make the signed sum of reciprocals up to n non-zero yet smaller than a constant over two to the n.
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.
The size of the largest subset of one through N carrying signs whose signed reciprocals sum to zero while no proper non-empty subset sums 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.
Estimates the number of ways an integer is a sum of k many kth powers, and asks whether it exceeds n to a fixed positive power infinitely often.
Asks whether the integers up to x that are sums of k kth powers number at least x to the power 1 minus epsilon, and whether sums of m such powers, m below k, number at least a constant times x to the m over k.
Asks whether some polynomial with integer coefficients has all sums of two of its values at distinct nonnegative integers distinct.
Asks whether the number of integers up to x that are sums of three nonnegative kth powers is at least a constant times x to the power three over k.