# Engel Expansion

The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that

Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. If x is rational, its Engel expansion provides a representation of x as an Egyptian fraction. Engel expansions are named after Friedrich Engel, who studied them in 1913.

An expansion analogous to an Engel expansion, in which alternating terms are negative, is called a Pierce expansion.

### Other articles related to "engel expansion, expansion, engel expansions, expansions":

Engel Expansion - Growth Rate of The Expansion Terms
... The coefficients ai of the Engel expansion typically exhibit exponential growth more precisely, for almost all numbers in the interval (0,1], the limit exists and is equal to e ... The same typical growth rate applies to the terms in expansion generated by the greedy algorithm for Egyptian fractions ... However, the set of real numbers in the interval (0,1] whose Engel expansions coincide with their greedy expansions has measure zero, and Hausdorff dimension 1/2 ...
Egyptian Fraction - Modern Number Theory
... see Erdős–Graham conjecture, Znám's problem, and Engel expansion ... of bounding the length or maximum denominator in Egyptian fraction representations, finding expansions of certain special forms or in which the denominators are all of some special type, the termination of ... represented using fewer fractions, as any expansion with repeated fractions can be converted to an Egyptian fraction of equal or smaller length by repeated application of the replacement if k ...

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