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Formulas and algorithms for the length of a Farey sequence
Vladimir Sukhoy1, Alexander Stoytchev2
1Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, 50011, USA.
Scientific Reports
|November 16, 2021
Summary
This study introduces new formulas and efficient algorithms for calculating the length of Farey sequences. The fastest algorithm achieves optimal time and memory complexity for computing Farey sequence lengths.
Area of Science:
- Number Theory
- Discrete Mathematics
- Computational Mathematics
Background:
- Farey sequences are fundamental in number theory.
- Efficient computation of Farey sequence length is crucial for various applications.
- Existing methods for computing Farey sequence length have limitations.
Purpose of the Study:
- To derive novel formulas for the length of a Farey sequence of order n.
- To develop and present iterative algorithms for computing Farey sequence length.
- To establish a new state-of-the-art algorithm for efficient Farey sequence length computation.
Main Methods:
- Derivation of new mathematical formulas for Farey sequence length.
- Development of iterative algorithms based on these formulas.
- Analysis of computational complexity (time and memory) for the algorithms.
Main Results:
- Several novel formulas for Farey sequence length are proven.
- A progression of iterative algorithms is presented, from slowest to fastest.
- The most efficient algorithm runs in O(n) time and uses O(1) memory.
- The length of Farey sequence of order 10^18 was computed.
Conclusions:
- The new formulas and algorithms significantly advance the computation of Farey sequence lengths.
- The developed algorithm is the most efficient to date.
- This work enables computation for extremely large orders of Farey sequences.
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