Applications of Minkowski’s geometry of numbers

Hengyuan Gan

doi:10.1117/12.2628216

22 April 2022 Applications of Minkowski’s geometry of numbers

Hengyuan Gan

Proceedings Volume 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021); 1216331 (2022) https://doi.org/10.1117/12.2628216
Event: International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 2021, Nanjing, China

Abstract

This work presents Minkowski’s theory of geometry of numbers. Theorem 1 Let M ⊂ Rn be a lattice, 𝐶 ⊂ ℝ^𝑛 be a subset. If Vol(𝐶) > Covol(𝑀), then there exists 𝑥 ≠ 𝑦 ∈ 𝐶 such that 𝑥 − 𝑦 ∈ 𝑀. We also use it to give congruence criteria for representability of forms 𝑎² + 𝑑𝑏²of prime numbers and prove Fermat’s four squares theorem. Theorem 2 Every positive integer is the sum of four squares.

Citation Download Citation

Hengyuan Gan "Applications of Minkowski’s geometry of numbers", Proc. SPIE 12163, International Conference on Statistics, Applied Mathematics, and Computing Science (CSAMCS 2021), 1216331 (22 April 2022); https://doi.org/10.1117/12.2628216

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