Extended Lagrange's four-square theorem

2018/05/11 by Lacalle, Jesús, Gatti, Laura N.
#11D09 #11H06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1805.04353

Abstract

Lagrange's four-square theorem states that every natural number n can be represented as the sum of four integer squares: n=x₁²+x₂²+x₃²+x₄². Ramanujan generalized Lagrange's result by providing, up to equivalence, all 54 quadratic forms ax₁²+bx₂²+cx₃²+dx₄² that represent all positive integers. In this article, we prove the following extension of Lagrange's theorem: given a prime number p and v₁∈ Z⁴, …, v_k∈ Z⁴, 1≤ k≤ 3, such that ‖v_i‖²=p for all 1≤ i≤ k and ⟨ v_i|v_j⟩=0 for all 1≤ i

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