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Multigrade Equation

A $(k,l)$-multigrade equation is a Diophantine Equation of the form

\sum_{i=1}^l n_i^j = \sum_{i=1}^l m_i^j

for $j=1$, ..., $k$, where m and n are $l$-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 $k=9$ and $l=10$ 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

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

See also Diophantine Equation


Chen, S. ``Equal Sums of Like Powers: On the Integer Solution of the Diophantine System.''

Gloden, A. Mehrgeradige Gleichungen. Groningen, Netherlands: Noordhoff, 1944.

Gloden, A. ``Sur la multigrade $A_1$, $A_2$, $A_3$, $A_4$, $A_5\mathrel{\mathop{=}\limits^k} B_1$, $B_2$, $B_3$, $B_4$, $B_5$ ($k=1$, 3, 5, 7).'' Revista Euclides 8, 383-384, 1948.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 143, 1994.

Kraitchik, M. ``Multigrade.'' §3.10 in Mathematical Recreations. New York: W. W. Norton, p. 79, 1942.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 171-173, 1979.

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