The golden ratio as a number base

The Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …) are one of the most famous sequences of integers. Every number in the sequence after the first two is the sum of the two previous numbers: 1+1=2, 1+2=3, 2+3=5, and so on. It turns out that every positive integer can be written as a sum of distinct Fibonacci numbers, possibly in several different ways. For example, 64 can be written as 34+21+8+1, or as 34+21+5+3+1, or as 34+13+8+5+3+1. All of these expressions involve at least two consecutive Fibonacci numbers, like 21 and 34, or 3 and 5.

According to Zeckendorf’s Theorem, every positive integer can be represented in a unique way as a sum of distinct, non-consecutive Fibonacci numbers. For example, the Zeckendorf representation of 64 is 55+8+1. There is a simple algorithm to find this representation, as follows. The largest Fibonacci number less than 64 is 55, so we subtract 55 from 64, which gives 9. The largest Fibonacci number less than 9 is 8, so we subtract 8 from 9, which gives 1. Since 1 is a Fibonacci number, the process terminates with 64=55+8+1.

The Fibonacci numbers are closely related to the golden ratio, (1+√5)/2, which is approximately 1.618. The golden ratio is the unique positive number φ with the property that φ²=φ+1. Even though the golden ratio is irrational, it can be used to give an exact formula for the n-th Fibonacci number: F_n=(φⁿ–(-φ^-n))/√5. For example, substituting n=3 and n=4 gives the third and fourth Fibonacci numbers, F₃=2 and F₄=3. Another connection between the Fibonacci numbers and the golden ratio is that the ratio of two consecutive Fibonacci numbers approaches the golden ratio as the numbers get large, which explains why 55/34 is approximately 1.618.

In 1957, George Bergman proved that every positive real number can be written as a sum of distinct powers of φ, where the powers of φ are allowed to be negative, and where infinite sums are allowed. In base 10, every terminating decimal has two representations, so that for example 1.53 may also be written as 1.52999 recurring, and something similar happens in every number base, including base φ. For example, φ itself can be expressed as the infinite sum 1 + φ^-2 + φ^-4 + φ^-6 + φ^-8 + …

The φ-representation of a number satisfies a uniqueness property reminiscent of Zeckendorf’s theorem. More precisely, the representation is unique, modulo the ambiguities of the previous paragraph, if we impose the additional condition that no consecutive powers of φ appear. For example, we would choose to represent the number φ²=φ¹+φ⁰ as φ² rather than as φ¹+φ⁰, in order to avoid consecutive powers of φ.

I found out about this particular theorem of Bergman from the recent paper New properties of the φ-representation of integers by Jeffrey Shallit and Ingrid Vukusic. The paper is concerned with integers k whose φ-representation is “φ-anti-palindromic”, which means that φⁿ appears in the φ-representation of k if and only if φ^-n also appears in the φ-representation of k. An example of such an integer is 25, whose φ-representation is φ⁶ + φ⁴ + φ^-4 + φ^-6. The set of all such integers was introduced by Vladimir Shevelev in 2010, and the first few terms are listed above.

In 2012, Clark Kimberling noticed that the numbers in Shevelev’s sequence seem to have the property that doubling all the exponents in the φ-representation produces another integer. Kimberling conjectured that these numbers are the only ones with this property. For example, 10 is an element of Shevelev’s sequence, and 10 is represented as φ⁴ + φ² + φ^-2 + φ^-4. Doubling all the exponents produces φ⁸ + φ⁴ + φ^-4 + φ^-8, which (surprisingly) is another integer, in this case 54. The main result in Shallit and Vukusic’s paper is that Kimberling’s conjecture is true.

Another sequence that is closely related to the Fibonacci sequence and the golden ratio is the sequence of Lucas numbers. As with the Fibonacci numbers, each Lucas number (other than the first two) is the sum of the two previous numbers, the difference being that the sequence starts with “2, 1”. The first few Lucas numbers are 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, … As with the Fibonacci numbers, the ratios of successive Lucas numbers tend to the golden ratio, which explains why 76/47 is close to 1.618.

Shallit and Vukusic used the open source automated theorem prover Walnut to find several connections between φ-representations of integers and the Lucas numbers. For example, they prove in their paper that the smallest exponent in the φ-representation of a positive integer n ≥ 2 must be an even number, –2i, and that in this case, n must lie between the (2i–1)-st and (2i+1)-st Lucas number. For example, the number 54 lies between the 7th Lucas number (which is 29) and the 9th Lucas number (which is 76). The theorem then implies that the smallest exponent in the φ-representation of 54 will be –8, because 8 is the even number between 7 and 9. This agrees with the observation that the φ-representation of 54 is φ⁸ + φ⁴ + φ^-4 + φ^-8.

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Picture credits and relevant links

The picture of the Fibonacci spiral is by Romain. It appears on Wikipedia’s entry on Fibonacci numbers.

The picture of the Lucas spiral is by The Nth User and Begoon. It appears on WIkipedia’s entry on Lucas numbers.

The other graphics are my own work.

Wikipedia also has entries on the golden ratio base, Zeckendorf’s Theorem, and George M. Bergman.

Shevelev’s sequence is sequence A178482 in The On-Line Encyclopedia of Integer Sequences.

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