* Warren Smith <warren.wds@gmail.com> [Feb 22. 2012 07:07]:

Your task is to find two disjoint sets (or multisets) {A1,A2,A3,...,Ak} and {B1,B2,B3,...,Bk} of integers with equal 0th, 1th, 2th,..., (k-1)th moments.

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k=6: ?I do not know of any example?

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keywords: Multigrades, Prouhet-Tarry-Escott problem Here is an old snippet from the pile that didn't make it into the fxtbook: ---------------------------------------------- \jjindex{multigrades} Let $a,\,a+1$ be any two successive numbers, then $a^0=(a+1)^0$ (wow!). Let $a,\,a+1,\,a+2,\,a+3$ any four successive numbers, then $a^1+(a+3)^1=(a+1)^1+(a+2)^1$ (still not really exciting). % We write the relations as \eq{prototypes}: % \begin{subequations} \begin{eqnarray} 0^0 & = & 1^0 \\ 0^1 + 3^1 & = & 1^1 + 2^1 \end{eqnarray} % We also have % \begin{eqnarray} \jjformula{rel:sum2proto} 0^2 + 3^2 + 5^2 + 6^2 & = & 1^2 + 2^2 + 4^2 + 7^2 \end{eqnarray} and {\small \begin{eqnarray} 0^3 + 3^3 + 5^3 + 6^3 + 9^3 + 10^3 + 12^3 + 15^3 \;=\; 1^3 + 2^3 + 4^3 + 7^3 + 8^3 + 11^3 + 13^3 + 14^3 \end{eqnarray} }% {\small \begin{eqnarray} 0^4 + 3^4 + 5^4 + 6^4 + 9^4 + 10^4 + 12^4 + 15^4 + 17^4 + 18^4 + 20^4 + 23^4 + 24^4 + 27^4 + 29^4 + 30^4 \;=\; \\ \nonumber \;=\; 1^4 + 2^4 + 4^4 + 7^4 + 8^4 + 11^4 + 13^4 + 14^4 + 16^4 + 19^4 + 21^4 + 22^4 + 25^4 + 26^4 + 28^4 + 31^4 \end{eqnarray} }% \end{subequations} % In general, for a fixed exponent $k>0$, let $s=2^{k+1}$ and partition the set $\{0,\,1,\,2,\,3,\,\ldots{},\,s-1\}$ into two sets $S_0,\,S_1$ so that $S_0$ contains all elements $j$ ($0\leq{}j<s$) with parity zero, $S_1$ the elements with parity one. % Then for $d\in{}\CC$, \begin{eqnarray} \sum_{a\in{}S_0}{\left(a+d\right)^k} & = & \sum_{b\in{}S_1}{\left(b+d\right)^k} \end{eqnarray} % An example, for $k=2$ we have $S_0=\{0,\,3,\,5,\,6\}$, $S_1=\{1,\,2,\,4,\,7\}$ and so \begin{eqnarray} (0+d)^2 + (3+d)^2 + (5+d)^2 + (6+d)^2 & = & (1+d)^2 + (2+d)^2 + (4+d)^2 + (7+d)^2\qquad%layout \end{eqnarray} % Compare to relation~\jjref{rel:sum2proto}. % The prototypes for the exponent $k$ are also valid for all positive integer exponents less than $k$ and the exponent zero. The equations given are special solutions of the so-called \jjterm{multigrades} or \jjterm{Prouhet-Tarry-Escott} problem: % Find two sets $A$ and $B$ such that for a given $k$ \begin{eqnarray} \sum_{a\in{}A}{a^j} & = & \sum_{b\in{}B}{b^j} \qquad{}j=0,1,2,\ldots,k \end{eqnarray} % Both sets have the same number $n$ of elements. This is a $(k,n)$-multigrade. % We found solutions of $(k,2^{k})$-multigrades with the special property that the union of both sets contains $2^k$ successive numbers. % The shift invariance holds again, it is a property of all multigrade solutions. Multiplying the symbolic entries in a prototype gives a valid prototype which again is shift invariant. This means that, for example, relation~\jjref{rel:sum2proto} can be interpreted as % ? eqm(x,k,m=1)=(2*(m*1+x)^k+(4*m+x)^k)-((0*m+x)^k+2*(3*m+x)^k) % ? for(x=-11,11,print1(" ",eqm(x,2,3))) % 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \begin{eqnarray} (0\,m+d)^k + (3\,m+d)^k + (5\,m+d)^k + (6\,m+d)^k \;=\; \\ \nonumber \;=\; (1\,m+d)^k + (2\,m+d)^k + (4\,m+d)^k + (7\,m+d)^k \end{eqnarray} for $k\in{}\{0,1,2\}$, $d\in{}\CC$ and $m\in{}\CC$. Adding and subtraction shifted versions of the prototypes gives more prototypes. We subtract from relation~\jjref{rel:sum2proto} the shifted by one version with swapped sides: % % \begin{eqnarray} \nonumber +\left(0^2 + 3^2 + 5^2 + 6^2\right. & = & \left.1^2 + 2^2 + 4^2 + 7^2\right) \\ \nonumber -\left(2^2 + 3^2 + 5^2 + 8^2\right. & = & \left.1^2 + 4^2 + 6^2 + 7^2\right) \end{eqnarray} % Giving $2\cdot{}2^2+8^2=0^2+2\cdot{}6^2$. We can divide all symbolic entries by two to obtain: \begin{eqnarray} 2\cdot{}1^2+4^2 & = & 0^2+2\cdot{}3^2 \end{eqnarray} which really is \begin{eqnarray} 2\cdot{}(1\,m+d)^k+(4\,m+d)^k & = & (0\,m+d)^k+2\cdot{}(3\,m+d)^k \end{eqnarray} for $k\in{}\{0,1,2\}$, $d\in{}\CC$ and $m\in{}\CC$. ---------------------------------------------- See Jean-Paul Allouche, Jeffrey Shallit: {The ubiquitous Prouhet-Thue-Morse sequence}, In: C.\ Ding, T.\ Helleseth, H.\ Niederreiter, (eds.), {Sequences and Their Applications}: Proceedings of SETA'98, pp.1-16, Springer-Verlag, (1999). %% http://www.cs.uwaterloo.ca/~shallit/papers.html