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Section: New Results

Stern-Brocot and Fibonacci sequences

Participant : José Grimm.

We constructed an explicit bijection 𝐍𝐐, first in the framework of the Bourbaki project (see above), then in Ssreflect. Every positive rational number x can uniquely be written as a quotient sn/sn+1. This result was established by Dijkstra who stated it in an obfuscated way. It was shown years before by Stern. It is possible to compute sn/sn+1 without computing numerator and denominator separately, by considering the sequences of bits of n from left to right or from right to left. Truncating the binary expansion of n yields a sequence of approximations to sn/sn+1 (this was studied by Brocot, and the so-called Stern-Brocot tree is an alternative representation of rational numbers). We implemented the work of Dijkstra and Stern in Coq [17] .

We also studied how a number can be represented by a sequence of other numbers (for instance as a sum of distinct Fibonacci numbers, with or without constraints). The number of ways of writing n as a sum of powers of two, each power of two being used at most twice, is sn+1. These results are presented in [17] .