Unit Fractions

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E0045/: Asks whether for each k some integer has its nontrivial divisors so arranged that any k-coloring leaves a monochromatic set of reciprocals summing to one.

E0046/: Asks whether every finite coloring of the integers admits a monochromatic set of distinct integers above one whose reciprocals sum to one.

E0047/: Asks whether a subset of the first N integers whose reciprocal sum exceeds a fixed multiple of the logarithm of N has a subset of reciprocals summing to one.

E0148/: Estimates the number of ways to write one as a sum of reciprocals of k distinct increasing positive integers.

E0206/: Asks whether, for almost every positive real number, the best sums of n distinct unit fractions below it are eventually built greedily.

E0242/: Asks whether every integer greater than 2 has four over it written as a sum of three reciprocals of distinct positive integers.

E0282/: Asks whether the greedy algorithm picking the least allowed denominator always terminates for a rational with odd denominator when only odd ones are allowed.

E0283/: 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.

E0284/: Asks whether the largest possible smallest denominator among k distinct unit fractions summing to one is asymptotically k divided by e minus one.

E0285/: Asks whether the least possible largest denominator among k distinct unit fractions summing to one is asymptotically e over e minus one, times k.

E0286/: 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.

E0287/: Asks whether distinct denominators above one whose unit fractions sum to one must always include two consecutive ones differing by at least three.

E0288/: Asks whether only finitely many pairs of integer intervals have their combined sum of reciprocals equal to a whole number.

E0289/: 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.

E0290/: 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.

E0291/: 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.

E0292/: Asks whether the integers that can occur as the largest denominator in a representation of one by distinct unit fractions have density one.

E0293/: 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.

E0294/: 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.

E0295/: 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.

E0296/: 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.

E0297/: Counts the subsets of the integers one through N whose reciprocals sum to one.

E0298/: Asks whether every set of positive integers of positive density contains a finite subset whose reciprocals sum to one.

E0299/: Asks whether some infinite increasing sequence of integers with bounded gaps has no finite subset of reciprocals summing to one.

E0300/: Estimates the size of the largest subset of one through N having no subset whose reciprocals sum to one.

E0301/: 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.

E0302/: 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.

E0303/: 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.

E0304/: 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.

E0305/: 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.

E0306/: Asks whether every positive rational with squarefree denominator is a sum of distinct unit fractions whose denominators are all products of two distinct primes.

E0307/: Asks whether two finite sets of primes exist whose sums of reciprocals multiply together to give one.

E0308/: 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.

E0309/: 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.

E0310/: 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.

E0311/: 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.

E0312/: 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.

E0313/: Asks whether infinitely many sums of reciprocals of distinct primes equal one minus the reciprocal of an integer.

E0314/: 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.

E0315/: 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.

E0316/: 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.

E0317/: 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.

E0318/: 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.

E0319/: 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.

E0320/: Estimates how many distinct values arise as sums of reciprocals of subsets of the integers one through N.

E0321/: The largest subset of the first N integers all of whose subsets have distinct sums of reciprocals.

E0327/: Asks how large a subset of the first N integers can be if the sum of any two distinct members never divides their product, or never divides twice it.

E0355/: Asks whether some sequence growing at least geometrically has finite sums of reciprocals of its terms covering every rational in some open interval.

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Egyptian fractions and reciprocal sums, including the Erdos-Straus conjecture and the Erdos-Graham problems on representing rationals as sums of unit fractions.

Site tags routed here: number theory, ramsey theory, unit fractions.