Universal sums of three quadratic polynomials

Abstract

Let a, b, c, d, e and f be integers with a ⩾ 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 ⩾ d if a = 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.