2024/11/21 by Zhi-Wei Sun, Sun, Zhi-Wei
Mathematics · Computer Science · #Statistical and numerical algorithms #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2411.14308
In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer n>1 can be written as w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2 with w,x,y,z∈\mathbb Z. Let a and b be integers with a>0, b>-a and gcd(a,b)=1. When 2\nmid ab, we show that any sufficiently large integer can be written as \fracw(aw+b)2+\fracx(ax+b)2+\fracy(ay+b)2+\fracz(az+b)2 with w,x,y,z nonnegative integers. When 2| a and 2\nmid b, we prove that any sufficiently large integer can be written as w(aw+b)+x(ax+b)+y(ay+b)+z(az+b) with w,x,y,z nonnegative integers.