Universal sums of three quadratic polynomials

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2020,（3）
• 页码范围: 501-520
• 页数: 20页
• 语种: 英文

摘要

Let a,b,c,d,e and f be integers witha ≥ c ≥ e ＞ 0,b ＞-a and b ≡ a (mod 2),d ＞-c and d≡c (mod 2),f ＞-e and f ≡ e (mod 2).Suppose that b ≥ difa =c,and d ≥ f if c =e.When b(a-b),d(c-d) and f(e-f) are not all zero,we prove that if each n ∈ N ={0,1,2,…} can be written as x(ax + b)/2 + y(cy + d)/2 + z(ez + f)/2 with x,y,z ∈ N then the tuple (a,b,c,d,e,f) must be on our list of 473 candidates,and show that 56 of them meet our purpose.When b ∈ [0,a),d ∈ [0,c) and f ∈ [0,e),we investigate the universal tuples (a,b,c,d,e,f) over Z for which any n ∈ N can be written as x(ax + b)/2 + y(cy + d)/2 + z(ez + f)/2 with x,y,z ∈ Z,and show that there are totally 12,082 such candidates some of which are proved to be universal tuples over Z.For example,we show that any n ∈ N can be written as x(x + 1)/2 + y(3y + 1)/2 + z(5z + 1)/2 with x,y,z ∈ Z,and conjecture that each n ∈ N can be written as x(x + 1)/2 + y(3y + 1)/2 + z(5z + 1)/2 with x,y,z ∈ N.