Engel expansion

The angels development of a positive real number x is monotonically increasing, unique sequence of natural numbers, so

Rational numbers have a finite Engel development, while it is infinite for irrational numbers. If x is rational, the angels development of x leads to an Egyptian fraction, that is a finite sum of reciprocals of natural numbers. The angels development was named after Friedrich Engel, who examined her in 1913.

Angel sequences, continued fractions and Fibonacci

Kraaikamp and Wu in 2004 observed that an angel - development can also be written as an ascending variant of a continued fraction:

They show that ascending continued fractions like these have been studied already in Fibonacci's Liber Abaci of 1202. This allows a connection between a general continued fraction development of the ascending continued fraction development and the angel - development can be made:

If all denominators are 0 or 1 in such a representation, as often occurs in the Liber Abaci, the result is an angel - development. The angels development has not yet been described as a general procedure of Fibonacci.

Algorithm for the calculation of angels developments

To find the angels development of x, set one

And write proceeds as follows:

Wherein the ceiling function (the smallest integer which is not less than r).

If for any i, finish execution.

Example

To find the angel Development of 1175, we perform the following steps:

The episode ends here. is therefore

And the angels development of 1,175 is {1, 6, 20}.

Angel developments of rational numbers

Every positive rational number has a unique finite Engel development. The algorithm is, if ui is a rational number x / y, ui 1 = (-y mod x) / y. The counter for the remaining fraction ui decreases and the construction of the Angel development must terminate in a finite number of steps. Every rational number also has a unique infinite angel Development: Using the identity

, the last digit of n in a finite Engel development be replaced without changing the value of the development through an infinite sequence of (n 1). for example

This fact is analogous to the fact that every rational number with a finite decimal representation also has an infinite decimal representation (eg 0.999 ... ).

Erdős, Rényi, and Szüsz asked for nontrivial limits on the length of a finite Engel development of a rational number x / y; This question was answered by Erdős and Shallit by proved that the number of terms in the expansion O ( y1 / 3 ε ) for each ε > 0.

Growth rate of the terms

The coefficients ai typically grow exponentially; more specifically, for almost all numbers in the interval ( 0,1] of the limit exists and is equivalent to E. The portion of the interval for which this is not the case, it is still large enough that its Hausdorff dimension is 1.

The same typical growth rate also applies for the terms in the greedy algorithm for Egyptian fractions. The share of real numbers in the interval ( 0,1], whose angel - development coincides with their greedy development, approaches 0, and the amount has Hausdorff dimension 1/2.

Other examples

Generally,

In general, an angel Development with constant terms is a geometric series. Other Engel developments for constants can be found here: .