On universal sums of polygonal numbers  

作　　者：SUN Zhi-Wei 

机构地区：[1]Department of Mathematics, Nanjing University

出　　处：《Science China Mathematics》2015年第7期1367-1396,共30页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11171140);the PAPD of Jiangsu Higher Education Institutions

摘　　要：For m = 3, 4,..., the polygonal numbers of order m are given by pm(n) =(m- 2) n2 + n(n= 0, 1, 2,...). For positive integers a, b, c and i, j, k 3 with max{i, j, k} 5, we call the triple(api, bpj, cpk)universal if for any n = 0, 1, 2,..., there are nonnegative integers x, y, z such that n = api(x) + bpj(y)+ cpk(z). We show that there are only 95 candidates for universal triples(two of which are(p4, p5, p6) and(p3, p4, p27)), and conjecture that they are indeed universal triples. For many triples(api, bpj, cpk)(including(p3, 4p4, p5),(p4, p5, p6) and(p4, p4, p5)), we prove that any nonnegative integer can be written in the form api(x) + bpj(y) + cpk(z) with x, y, z ∈ Z. We also show some related new results on ternary quadratic forms,one of which states that any nonnegative integer n ≡ 1(mod 6) can be written in the form x2+ 3y2+ 24z2 with x, y, z ∈ Z. In addition, we pose several related conjectures one of which states that for any m = 3, 4,...each natural number can be expressed as pm+1(x1) + pm+2(x2) + pm+3(x3) + r with x1, x2, x3 ∈ {0, 1, 2,...}and r ∈ {0,..., m- 3}.For m = 3, 4,..., the polygonal numbers of order m are given by pm（n） =（m- 2） n2 ＋ n（n= 0, 1, 2,...）. For positive integers a, b, c and i, j, k 3 with max{i, j, k} 5, we call the triple（api, bpj, cpk）universal if for any n = 0, 1, 2,..., there are nonnegative integers x, y, z such that n = api（x） ＋ bpj（y）＋ cpk（z）. We show that there are only 95 candidates for universal triples（two of which are（p4, p5, p6） and（p3, p4, p27））, and conjecture that they are indeed universal triples. For many triples（api, bpj, cpk）（including（p3, 4p4, p5）,（p4, p5, p6） and（p4, p4, p5））, we prove that any nonnegative integer can be written in the form api（x） ＋ bpj（y） ＋ cpk（z） with x, y, z ∈ Z. We also show some related new results on ternary quadratic forms,one of which states that any nonnegative integer n ≡ 1（mod 6） can be written in the form x2＋ 3y2＋ 24z2 with x, y, z ∈ Z. In addition, we pose several related conjectures one of which states that for any m = 3, 4,...each natural number can be expressed as pm＋1（x1） ＋ pm＋2（x2） ＋ pm＋3（x3） ＋ r with x1, x2, x3 ∈ {0, 1, 2,...}and r ∈ {0,..., m- 3}.

关 键 词：polygonal numbers ternary quadratic forms representations of integers PRIMES 

分 类 号：O156[理学—数学] O178[理学—基础数学]