On difference of two sums of two squares plus difference of two cubes

Question

Is there a way to show that every integer $n$ can be represented by
$(a^3+b^2+c^2)-(d^3+e^2+f^2)$
Where $a,b,c,d,e,f$ are all integers?

Answer 1

Yes, every integer $k$ is of this form. This is trivial, because we may take $a=d$ and some of the $b,c,e,f$ equal to zero. Every $k$ is either of the form $2n+1$ or $2n$. We have \begin{align*} 2n+1 & = (n+1)^2-n^2+0^2 \\ 2n & =(n+1)^2-n^2-1^2 \end{align*}

Answer 2

I have found that every integer $n$ can be represented by
$(a^2+b^2)-(c^2+d^2)$
Because $$n=((\frac{(n+1)(n+2)}{2})^2 + (\frac{n(n+1)(n+2)}{2})^2)-\left((\frac{n(n+1)(n+2)}{2}+1)^2 + (\frac{n(n+1)}{2})^2\right)$$
So, I am just wondering if there is at least an expression which suits the question i asked.

Answer 3

Actually, every integer can be expressed as $$ b^2 + c^2 - d^3 $$

This was a problem Kap and Noam Elkies gave to the MAA Monthly, Problems and Solutions column. It appeared in January 1995, Volume 102, number 1, page 70. The answers appeared in June-July 1997, Volume 104, number 6, page 574.

However, not every number can be expressed as $ b^2 + c^2 - d^9 $

Much later, we found that not every integer can be expressed as $$ 4 x^2 + 2 xy + 7 y^2 - z^3 $$

I sent that one into the Monthly, the question appeared in December 2010. The single answer, by Robin Chapman, appeared in December 2012, volume 119, number 10, pages 884-885

For the curious, see the more difficult https://mathoverflow.net/questions/12486/integers-not-represented-by-2-x2-x-y-3-y2-z3-z