Erdős–Graham problem

The Erdős–Graham problem asks whether every set of integers greater than one whose reciprocals have divergent sum contains a finite subset with reciprocal sum exactly one.

Connect

Egyptian fraction: A representation of a rational number as a sum of distinct unit fractions. The target equation is an Egyptian-fraction representation of one using denominators from the given set.
Paul Erdős: A Hungarian mathematician known for prolific collaboration and major contributions to number theory and combinatorics. Erdős posed the problem jointly with Graham as a question about reciprocal sums.
Reciprocal: For a nonzero number x, its reciprocal is the number 1/x. Each chosen integer contributes its reciprocal to the sum tested by the problem.
Practical numbers: Positive integers for which every smaller positive integer is a sum of distinct divisors. Their reciprocal sums diverge, making them a natural structured test set for the problem.
Erdős–Graham theorem: A theorem asserting that every set of integers greater than one with reciprocal sum exceeding a universal constant contains a finite subset summing reciprocally to one. The theorem proves a stronger-than-local result under a fixed large-sum bound, but not the full divergent-sum conjecture.
Harmonic series: The divergent series formed by summing the reciprocals of all positive integers. The problem generalizes the harmonic series by restricting denominators to an arbitrary set.
Ronald Graham: An American mathematician whose work ranged across combinatorics, number theory, and recreational mathematics. Graham co-formulated the problem with Erdős.
Subset: A set whose elements all belong to another set. The desired unit sum must use a finite selection of distinct members of the original set.
Arithmetic progression: A sequence of numbers with a constant difference between consecutive terms. Sets defined by congruence classes or progressions provide structured examples for reciprocal-sum questions.
Erdős–Graham constant: A universal reciprocal-sum threshold sufficient to guarantee a finite subset whose reciprocals sum to one. The threshold quantifies the proved guarantee and highlights the gap from mere divergence.