Representations Of Integers And Problems On Complete Systems Of Residues Of Polygonal Numbers Modulo K

In 2017,Z-W Sun conjectured that for any positive integers a,b,c,d,e,we have{x4+ ay4 + bz4 + cu4 + dv4 + ew4:x,y,z,u,v,w ∈N } ≠ N,where N = {0,1,2,…}.We confirm this conjecture via computation and case-by-case discussion.For m = 1,2,3,…,the polygonal numbers of order m + 2 are those Pm+2(n)=m(2n)+n(n =0,1,2,…).We determine completely those positive integers k such that {Pm+2(n):n=0,1,2,…} contains a complete system of residues modulo k.