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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.
According to Zeckendorf’s Theorem, every positive integer can be represented in a unique way as a sum of distinct, non-consecutive Fibonacci numbers. 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. Shallit and Vukusic used the open source automated theorem prover Walnut to find several connections between φ-representations of integers and the Lucas numbers.
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