Multigrade Equation

A -multigrade equation is a Diophantine Equation of the form

$\begin{displaymath} \sum_{i=1}^l n_i^j = \sum_{i=1}^l m_i^j \end{displaymath}$

for

, ...,

, where m and n are

-Vectors. Multigrade identities remain valid if a constant is added to each element of m and n (Madachy 1979), so multigrades can always be put in a form where the minimum component of one of the vectors is 1.

Small-order examples are the (2, 3)-multigrade with ${\bf m}=\{1,6,8\}$ and ${\bf n}=\{2,4,9\}$ :

$\displaystyle \sum_{i=1}^3 m_i^1$	$\textstyle =$	$\displaystyle \sum_{i=1}^3 n_i^1=15$
$\displaystyle \sum_{i=1}^3 m_i^2$	$\textstyle =$	$\displaystyle \sum_{i=1}^3 n_i^2=101,$

the (3, 4)-multigrade with ${\bf m}=\{1, 5, 8, 12\}$ and ${\bf n}=\{2, 3, 10, 11\}$ :

$\displaystyle \sum_{i=1}^4 m_i^1$	$\textstyle =$	$\displaystyle \sum_{i=1}^4 n_i^1=26$
$\displaystyle \sum_{i=1}^4 m_i^2$	$\textstyle =$	$\displaystyle \sum_{i=1}^4 n_i^2=234$
$\displaystyle \sum_{i=1}^4 m_i^3$	$\textstyle =$	$\displaystyle \sum_{i=1}^4 n_i^3=2366,$

and the (4, 6)-multigrade with ${\bf m}=\{1, 5, 8, 12, 18, 19\}$ and ${\bf n}=\{2, 3, 9, 13, 16, 20\}$ :

$\displaystyle \sum_{i=1}^6 m_i^1$	$\textstyle =$	$\displaystyle \sum_{i=1}^6 n_i^1=63$
$\displaystyle \sum_{i=1}^6 m_i^2$	$\textstyle =$	$\displaystyle \sum_{i=1}^6 n_i^2=919$
$\displaystyle \sum_{i=1}^6 m_i^3$	$\textstyle =$	$\displaystyle \sum_{i=1}^6 n_i^3=15057$
$\displaystyle \sum_{i=1}^6 m_i^4$	$\textstyle =$	$\displaystyle \sum_{i=1}^6 n_i^4=260755$

(Madachy 1979).

A spectacular example with and is given by ${\bf n}=\{\pm 12, \pm 11881, \pm 20231, \pm 20885, \pm 23738\}$ and ${\bf m}=\{\pm 436, \pm 11857, \pm 20449, \pm 20667, \pm 23750\}$ (Guy 1994), which has sums

$\displaystyle \sum_{i=1}^9 m_i^1$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^1=0$
$\displaystyle \sum_{i=1}^9 m_i^2$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^2=3100255070$
$\displaystyle \sum_{i=1}^9 m_i^3$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^3=0$
$\displaystyle \sum_{i=1}^9 m_i^4$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^4=1390452894778220678$
$\displaystyle \sum_{i=1}^9 m_i^5$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^5=0$
$\displaystyle \sum_{i=1}^9 m_i^6$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^6=666573454337853049941719510$
$\displaystyle \sum_{i=1}^9 m_i^7$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^7=0$
$\displaystyle \sum_{i=1}^9 m_i^8$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^8=330958142560259813821203262692838598$
$\displaystyle \sum_{i=1}^9 m_i^9$	$\textstyle =$	$\displaystyle \sum_{i=1}^9 n_i^9=0.$