# Engel expansion

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

For instance, Euler's constant *e* has the Engel expansion^{[1]}

- 1, 1, 2, 3, 4, 5, 6, 7, 8, ...

corresponding to the infinite series

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.

## Contents

- 1 Engel expansions, continued fractions, and Fibonacci
- 2 Algorithm for computing Engel expansions
- 3 Iterated functions for computing Engel expansions
- 4 Relation to the Riemann function
- 5 Example
- 6 Engel expansions of rational numbers
- 7 Engel expansions for some well-known constants
- 8 Growth rate of the expansion terms
- 9 Notes
- 10 References
- 11 External links

## Engel expansions, continued fractions, and Fibonacci[edit]

Kraaikamp & Wu (2004) observe that an Engel expansion can also be written as an ascending variant of a continued fraction:

They claim that ascending continued fractions such as this have been studied as early as Fibonacci's *Liber Abaci* (1202); this claim appears to refer to Fibonacci's compound fraction notation in which a sequence of numerators and denominators sharing the same fraction bar represents an ascending continued fraction:

If such a notation has all numerators 0 or 1, as occurs in several instances in *Liber Abaci*, the result is an Engel expansion. However, Engel expansion as a general technique does not seem to be described by Fibonacci.

## Algorithm for computing Engel expansions[edit]

To find the Engel expansion of *x*, let

and

where is the ceiling function (the smallest integer not less than *r*).

If for any *i*, halt the algorithm.

## Iterated functions for computing Engel expansions[edit]

Another equivalent method is to consider the map ^{[2]}

and set

where

- and

Yet another equivalent method, called the modified Engel expansion calculated by

and

### The Transfer operator of the Engel map[edit]

The Frobenius-Perron Transfer operator of the Engel map acts on functions with

since

and the inverse of the n-th component is which is found by solving for .

## Relation to the Riemann function[edit]

The Mellin transform of the map is related to the Riemann zeta function by the formula

## Example[edit]

To find the Engel expansion of 1.175, we perform the following steps.

The series ends here. Thus,

and the Engel expansion of 1.175 is {1, 6, 20}.

## Engel expansions of rational numbers[edit]

Every positive rational number has a unique finite Engel expansion. In the algorithm for Engel expansion, if *u _{i}* is a rational number

*x*/

*y*, then

*u*

_{i+1}= (−

*y*mod

*x*)/

*y*. Therefore, at each step, the numerator in the remaining fraction

*u*decreases and the process of constructing the Engel expansion must terminate in a finite number of steps. Every rational number also has a unique infinite Engel expansion: using the identity

_{i}the final digit *n* in a finite Engel expansion can be replaced by an infinite sequence of (*n* + 1)s without changing its value. For example,

This is analogous to the fact that any rational number with a finite decimal representation also has an infinite decimal representation (see 0.999...). An infinite Engel expansion in which all terms are equal is a geometric series.

Erdős, Rényi, and Szüsz asked for nontrivial bounds on the length of the finite Engel expansion of a rational number *x*/*y*; this question was answered by Erdős and Shallit, who proved that the number of terms in the expansion is O(*y*^{1/3 + ε}) for any ε > 0.^{[3]}

## Engel expansions for some well-known constants[edit]

And in general,

More Engel expansions for constants can be found here.

## Growth rate of the expansion terms[edit]

The coefficients *a _{i}* 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*. However, the subset of the interval for which this is not the case is still large enough that its Hausdorff dimension is one.

^{[4]}

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.^{[5]}

## Notes[edit]

**^**Sloane, N. J. A. (ed.). "Sequence A028310".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**Sloane, N. J. A. (ed.). "Sequence A220335".*The On-Line Encyclopedia of Integer Sequences*. OEIS Foundation.**^**Erdős, Rényi & Szüsz (1958); Erdős & Shallit (1991).**^**Wu (2000). Wu credits the result that the limit is almost always*e*to Janos Galambos.**^**Wu (2003).

## References[edit]

- Engel, F. (1913), "Entwicklung der Zahlen nach Stammbruechen",
*Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg*, pp. 190–191. - Pierce, T. A. (1929), "On an algorithm and its use in approximating roots of algebraic equations",
*American Mathematical Monthly*,**36**(10): 523–525, doi:10.2307/2299963, JSTOR 2299963 - Erdős, Paul; Rényi, Alfréd; Szüsz, Peter (1958), "On Engel's and Sylvester's series" (PDF),
*Ann. Univ. Sci. Budapest. Eötvös Sect. Math.*,**1**: 7–32. - Erdős, Paul; Shallit, Jeffrey (1991), "New bounds on the length of finite Pierce and Engel series",
*Journal de théorie des nombres de Bordeaux*,**3**(1): 43–53, doi:10.5802/jtnb.41, MR 1116100. - Paradis, J.; Viader, P.; Bibiloni, L. (1998), "Approximation to quadratic irrationals and their Pierce expansions",
*Fibonacci Quarterly*,**36**(2): 146–153 - Kraaikamp, Cor; Wu, Jun (2004), "On a new continued fraction expansion with non-decreasing partial quotients",
*Monatshefte für Mathematik*,**143**(4): 285–298, doi:10.1007/s00605-004-0246-3. - Wu, Jun (2000), "A problem of Galambos on Engel expansions",
*Acta Arithmetica*,**92**(4): 383–386, MR 1760244. - Wu, Jun (2003), "How many points have the same Engel and Sylvester expansions?",
*Journal of Number Theory*,**103**(1): 16–26, doi:10.1016/S0022-314X(03)00017-9, MR 2008063.

## External links[edit]

- Weisstein, Eric W. "Engel Expansion". MathWorld–A Wolfram Web Resource.