Finding the smallest positive integers such that $sum_{k=1}^{n} a_{k}^{r} = sum_{k=1}^{n} b_{k}^{r}$, for all...
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So we are trying to find two sets of positive integers $lbrace a_{1},...,a_{n} rbrace$ and $lbrace b_{1},...,b_{n}rbrace$, with nomember of one set in the other, such that $$sum_{k=1}^{n} a_{k} = sum_{k=1}^{n} b_{k}$$$$sum_{k=1}^{n} a_{k}^{2} = sum_{k=1}^{n} b_{k}^{2}$$
$$vdots$$
$$sum_{k=1}^{n} a_{k}^{N} = sum_{k=1}^{n} b_{k}^{N}$$
For some integer $r$. I define 'smallest' in the question title as the solution set with smallest value of $sum_{k=1}^{n} a_{k}$.
For example, for $N=1$ the solution is, $lbrace 3,1rbrace$ and $lbrace 2,2 rbrace$ as there is only one way to sum $2$ and $3$. For $N=2$, I believe the smallest solution is $lbrace 4,4,1rbrace$ and $lbrace 5,2,2 rbrace$. How can I find solutions for higher values of $N$?
number-theory
asked Nov 22 at 11:33
Joshua Farrell
600218
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0
down vote
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So we are trying to find two sets of positive integers $lbrace a_{1},...,a_{n} rbrace$ and $lbrace b_{1},...,b_{n}rbrace$, with nomember of one set in the other, such that $$sum_{k=1}^{n} a_{k} = sum_{k=1}^{n} b_{k}$$$$sum_{k=1}^{n} a_{k}^{2} = sum_{k=1}^{n} b_{k}^{2}$$
$$vdots$$
$$sum_{k=1}^{n} a_{k}^{N} = sum_{k=1}^{n} b_{k}^{N}$$
For some integer $r$. I define 'smallest' in the question title as the solution set with smallest value of $sum_{k=1}^{n} a_{k}$.
For example, for $N=1$ the solution is, $lbrace 3,1rbrace$ and $lbrace 2,2 rbrace$ as there is only one way to sum $2$ and $3$. For $N=2$, I believe the smallest solution is $lbrace 4,4,1rbrace$ and $lbrace 5,2,2 rbrace$. How can I find solutions for higher values of $N$?
number-theory
asked Nov 22 at 11:33
Joshua Farrell
600218
What do you mean by "with member of one set in the other"? Also, do solutions exist for all $N$?
– Servaes
Nov 22 at 12:07
In its full generality, this is a notorious unsolved problem that goes by the name of Tarry-Escott, or multigrades. Those search terms will get you many hits on the net in general and on MathOverflow and math.stackexchange in particular.
– Gerry Myerson
Nov 22 at 12:09
add a comment |
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0
down vote
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up vote
0
down vote
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So we are trying to find two sets of positive integers $lbrace a_{1},...,a_{n} rbrace$ and $lbrace b_{1},...,b_{n}rbrace$, with nomember of one set in the other, such that $$sum_{k=1}^{n} a_{k} = sum_{k=1}^{n} b_{k}$$$$sum_{k=1}^{n} a_{k}^{2} = sum_{k=1}^{n} b_{k}^{2}$$
$$vdots$$
$$sum_{k=1}^{n} a_{k}^{N} = sum_{k=1}^{n} b_{k}^{N}$$
For some integer $r$. I define 'smallest' in the question title as the solution set with smallest value of $sum_{k=1}^{n} a_{k}$.
For example, for $N=1$ the solution is, $lbrace 3,1rbrace$ and $lbrace 2,2 rbrace$ as there is only one way to sum $2$ and $3$. For $N=2$, I believe the smallest solution is $lbrace 4,4,1rbrace$ and $lbrace 5,2,2 rbrace$. How can I find solutions for higher values of $N$?
number-theory
asked Nov 22 at 11:33
Joshua Farrell
600218
So we are trying to find two sets of positive integers $lbrace a_{1},...,a_{n} rbrace$ and $lbrace b_{1},...,b_{n}rbrace$, with nomember of one set in the other, such that $$sum_{k=1}^{n} a_{k} = sum_{k=1}^{n} b_{k}$$$$sum_{k=1}^{n} a_{k}^{2} = sum_{k=1}^{n} b_{k}^{2}$$
$$vdots$$
$$sum_{k=1}^{n} a_{k}^{N} = sum_{k=1}^{n} b_{k}^{N}$$
For some integer $r$. I define 'smallest' in the question title as the solution set with smallest value of $sum_{k=1}^{n} a_{k}$.
For example, for $N=1$ the solution is, $lbrace 3,1rbrace$ and $lbrace 2,2 rbrace$ as there is only one way to sum $2$ and $3$. For $N=2$, I believe the smallest solution is $lbrace 4,4,1rbrace$ and $lbrace 5,2,2 rbrace$. How can I find solutions for higher values of $N$?
number-theory
number-theory
asked Nov 22 at 11:33
Joshua Farrell
600218
asked Nov 22 at 11:33
Joshua Farrell
600218
edited Nov 22 at 20:22
asked Nov 22 at 11:33
Joshua Farrell
600218
asked Nov 22 at 11:33
Joshua Farrell
600218
asked Nov 22 at 11:33
Joshua Farrell
600218
600218
What do you mean by "with member of one set in the other"? Also, do solutions exist for all $N$?
– Servaes
Nov 22 at 12:07
In its full generality, this is a notorious unsolved problem that goes by the name of Tarry-Escott, or multigrades. Those search terms will get you many hits on the net in general and on MathOverflow and math.stackexchange in particular.
– Gerry Myerson
Nov 22 at 12:09
add a comment |
What do you mean by "with member of one set in the other"? Also, do solutions exist for all $N$?
– Servaes
Nov 22 at 12:07
In its full generality, this is a notorious unsolved problem that goes by the name of Tarry-Escott, or multigrades. Those search terms will get you many hits on the net in general and on MathOverflow and math.stackexchange in particular.
– Gerry Myerson
Nov 22 at 12:09
What do you mean by "with member of one set in the other"? Also, do solutions exist for all $N$?
– Servaes
Nov 22 at 12:07
What do you mean by "with member of one set in the other"? Also, do solutions exist for all $N$?
– Servaes
Nov 22 at 12:07
In its full generality, this is a notorious unsolved problem that goes by the name of Tarry-Escott, or multigrades. Those search terms will get you many hits on the net in general and on MathOverflow and math.stackexchange in particular.
– Gerry Myerson
Nov 22 at 12:09
In its full generality, this is a notorious unsolved problem that goes by the name of Tarry-Escott, or multigrades. Those search terms will get you many hits on the net in general and on MathOverflow and math.stackexchange in particular.
– Gerry Myerson
Nov 22 at 12:09
add a comment |
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What do you mean by "with member of one set in the other"? Also, do solutions exist for all $N$?
– Servaes
Nov 22 at 12:07
In its full generality, this is a notorious unsolved problem that goes by the name of Tarry-Escott, or multigrades. Those search terms will get you many hits on the net in general and on MathOverflow and math.stackexchange in particular.
– Gerry Myerson
Nov 22 at 12:09