Universal sums of three quadratic polynomials

Abstract

Let $a,b,c,d,e$ and $f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $b\equiv a(\mathrm{mod}2)$, $d>-c$ and $d\equiv c(\mathrm{mod}2)$, $f>-e$ and $f\equiv e(\mathrm{mod}2)$. Suppose that $b\ge d$ if $a=c$, and $d\ge f$ if $c=e$. When $b(a-b)$, $d(c-d)$ and $f(e-f)$ are not all zero, we prove that if each $n\in \mathbb{N}=\{0,1,2,\dots \}$ can be written $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in \mathbb{N}$ 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 $b\in [0,a)$, $d\in [0,c)$ and $f\in [0,e)$, we investigate the universal tuples $(a,b,c,d,e,f)$ over $\mathbb{Z}$ for which any $n\in \mathbb{N}$ can be written $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\in \mathbb{Z}$, and show that there are totally 12082 such candidates some of which are proved to be universal tuples over $\mathbb{Z}$. For example, we show that any $n\in \mathbb{N}$ can be written as $x(x+1)/2+y(3y+1)/2+z(5z+1)/2$ with $x,y,z\in \mathbb{Z}$, and conjecture that each $n\in \mathbb{N}$ can be written as $x(x+1)/2+y(3y+1)/2+z(5z+1)/2$ with $x,y,z\in \mathbb{N}$.

26 pages

References in corpus (2)

• Ternary universal sums of generalized polygonal numbers
• Natural numbers represented by $\lfloor x^2/a\rfloor +\lfloor y^2/b\rfloor +\lfloor z^2/c\rfloor$

Cited by in corpus (5)

• Arithmetic progressions represented by diagonal ternary quadratic forms
• Some universal quadratic sums over the integers
• On sums of four pentagonal numbers with coefficients
• Sums of squares with restrictions involving primes
• Almost universal mixed sums of squares and polygonal numbers