Egyptian fraction

An Egyptian fraction expresses a positive rational number as a sum of distinct unit fractions, each with numerator 1.

Unit fraction: A fraction whose numerator is 1 and whose denominator is a positive integer. Every term in an Egyptian fraction is a unit fraction.
Rhind Mathematical Papyrus: An ancient Egyptian mathematical papyrus containing arithmetic problems and tables of fraction decompositions. Its unit-fraction table is a major surviving source for ancient Egyptian fraction practice.
Erdős–Straus conjecture: The conjecture that every integer n ≥ 2 has 4/n expressible as a sum of three unit fractions. It asks for a tightly bounded Egyptian fraction representation for every eligible integer.
Continued fraction: A nested expression formed from integers and reciprocals, used to represent rational and irrational numbers. Continued fractions use nested reciprocals rather than sums of distinct unit fractions.
Greedy algorithm for Egyptian fractions: A procedure that repeatedly subtracts the largest unit fraction not exceeding a given positive rational number. It produces an Egyptian fraction by choosing each denominator successively.
Ahmes: An Egyptian scribe who copied the mathematical text now known as the Rhind Mathematical Papyrus. The papyrus's fraction table is introduced as a work of Ahmes.
Diophantine equation: An equation whose solutions are sought among integers or other specified rational values. Finding Egyptian fraction representations is a Diophantine problem with denominator constraints.
Farey sequence: The ordered list of reduced fractions between zero and one whose denominators do not exceed a fixed bound. Farey sequences organize rational numbers by denominator, not by additive unit-fraction decomposition.
Fibonacci–Sylvester expansion: An Egyptian fraction expansion obtained by repeatedly taking the ceiling of the reciprocal of the remaining fraction. This named expansion is the standard output of the greedy method.
Egyptian hieratic numerals: A cursive writing system used in ancient Egypt for administrative and literary texts, including mathematical manuscripts. Ancient fraction decompositions were recorded in the hieratic mathematical tradition.

Linked from 7 pages