2024/08/21 by Ju, Jangwon
#11E12 #11E20 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2408.11310
For an arbitrary integer x, an integer of the form T(x) = (x2+x)/(2) is called a triangular number. Let α1,…,αk be positive integers. A sum Δα1,…,αk(x1,…,xk)=α1 T(x1)+⋯+αk T(xk) of triangular numbers is said to be even universal if the Diophantine equation Δα1,…,αk(x1,…,xk)=2n has an integer solution (x1,…,xk)∈ℤk for any nonnegative integer n. In this article, we classify all even universal sums of triangular numbers. Furthermore, we provide an effective criterion on even universality of an arbitrary sum of triangular numbers, which is a generalization of the triangular theorem of eight of Bosma and Kane.