Erdős–Straus conjecture

The Erdős–Straus conjecture states that every integer n ≥ 2 admits positive integers a, b, and c satisfying 4/n = 1/a + 1/b + 1/c. It remains unproved.

Egyptian fraction: A representation of a rational number as a sum of distinct unit fractions. The conjecture asks whether every 4/n has a representation using three positive unit fractions.
Reciprocal: The multiplicative inverse of a nonzero number; the reciprocal of x is 1/x. The equation expresses 4/n as a sum of reciprocals of integers.
Paul Erdős: A Hungarian mathematician known for prolific collaboration and contributions across number theory and combinatorics. Erdős posed the conjecture jointly with Ernst G. Straus.
Computational number theory: The use of algorithms and computation to investigate problems about integers. Computers have verified the identity for large finite ranges of n, but cannot alone settle all n.
Unit fraction: A rational number whose numerator is one and whose denominator is a positive integer. Each term in the conjecture’s proposed decomposition is a unit fraction.
Rational number: A number expressible as the quotient of two integers with nonzero denominator. Both sides of every proposed identity are rational numbers.
Ernst G. Straus: A mathematician whose work included number theory, geometry, and mathematical physics. Straus formulated the conjecture with Erdős.
Proof by infinite descent: A proof method showing that a supposed counterexample would produce a smaller one indefinitely. An infinite descent would rule out counterexamples if a suitable reduction were found.
Divisibility: The relation in which one integer is an exact multiple of another. Many known constructions split into cases according to divisibility conditions on n.
Prime number: An integer greater than one whose only positive divisors are one and itself. Prime inputs form a central subproblem in attempts to prove the conjecture.

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Erdős–Graham problemCompared with: Both ask for restricted unit-fraction representations, but this conjecture fixes the number being represented rather than a divergent denominator set.