On restricted sums of four squares and Zhi-Wei Sun's x+24y conjecture

2025/11/28 by Wu, Hai-Liang, She, Yue-Feng
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.23223

Abstract

In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange's four-square theorem. For example, given positive integers a,b satisfying some algebraic conditions and a positive integer C≥3, we will show that for any sufficiently large integer n with \ord₂(n)≤ C, there exist non-negative integers x,y,z,w such that \begincases x²+y²+z²+w²=n, ax+by\inS, \endcases where S is the set of all squares over ℤ. In particular, we obtain some progress on Zhi-Wei Sun's x+24y conjecture.

Citations

• Sums of four polygonal numbers: precise formulas
• Sums of four rational squares with certain restrictions
• Report on Zhi-Wei Sun's "1-3-5 conjecture" and some of its refinements
• Zhi-Wei Sun's 1-3-5 Conjecture and Variations
• Quaternary Quadratic Forms Representing all Integers

Related

• Sums of four squares with a certain restriction
• Sums of squares with restrictions involving primes
• Sums of four rational squares with certain restrictions
• On a Conjecture of Sun about sums of restricted squares
• Extended Lagrange's four-square theorem