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Number Theory
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Recurrent sequences
Linear recurrent sequences
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Lucas’ sequences
Fibonacci Numbers
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Zeckendorf's theorem states that every positive integer can be uniquely represented way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers, that is
Using this representation we can define a new product of two integers [1] : If and are two positive integers then their Knuth (also called Fibonacci) product is defined by
Knuth proved that this product is associative. If we change the definition in such a way that we define
then this product is not associative. On the other hand, 1 is a multiplicative identity in product but not in the product.
[1] | Knuth, D. E. (1988). Fibonacci multiplication. Appl. Math. Lett., 1, 57-60. |
Cite this web-page as:
Štefan Porubský: Knuth multiplication.