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Generalized Pythagorean integer solutions
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We know that $a^2 + b^2 = c^2$ has infinitely integer solutions which can be written as 3-tuples for example (3,4,5).
Can we conjecture that $$x_1^n +x_2^n + cdots +x_n^n = y^n$$ has infinitely many integer solutions for each $n in mathbb N$? If so how would one prove this to be correct?
I apologize if this is a simple question. I havent taken number theory but this is something that has caught my curiosity. Thanks in advance.
linear-algebra number-theory elementary-number-theory proof-verification recreational-mathematics
asked Jul 26 at 22:45
Red
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up vote
3
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We know that $a^2 + b^2 = c^2$ has infinitely integer solutions which can be written as 3-tuples for example (3,4,5).
Can we conjecture that $$x_1^n +x_2^n + cdots +x_n^n = y^n$$ has infinitely many integer solutions for each $n in mathbb N$? If so how would one prove this to be correct?
I apologize if this is a simple question. I havent taken number theory but this is something that has caught my curiosity. Thanks in advance.
linear-algebra number-theory elementary-number-theory proof-verification recreational-mathematics
asked Jul 26 at 22:45
Red
1,747733
1
related: mathoverflow.net/questions/62820/pythagorean-5-tuples
– Michael McGovern
Jul 26 at 22:52
1
An answer to this would follow from solutions to Waring's problem.
– Somos
Jul 27 at 0:00
Waring's problem is about the worst case, e.g., the number of 6th powers necessary to express each number as a sum. That number will be greater than 6, so solving Waring won't tell you anything about writing a 6th power as a sum of 6 6th powers.
– Gerry Myerson
Jul 27 at 7:03
1
But, Red, you might want to insist on positive integer solutions, else you can have $x_1=y$, $x_2=cdots=x_n=0$.
– Gerry Myerson
Jul 27 at 7:04
According to en.wikipedia.org/wiki/Sixth_power#Sums "no examples are yet known of a sixth power expressible as the sum of just six sixth powers."
– Gerry Myerson
Jul 27 at 7:07
 |Â
show 1 more comment
up vote
3
down vote
favorite
up vote
3
down vote
favorite
We know that $a^2 + b^2 = c^2$ has infinitely integer solutions which can be written as 3-tuples for example (3,4,5).
Can we conjecture that $$x_1^n +x_2^n + cdots +x_n^n = y^n$$ has infinitely many integer solutions for each $n in mathbb N$? If so how would one prove this to be correct?
I apologize if this is a simple question. I havent taken number theory but this is something that has caught my curiosity. Thanks in advance.
linear-algebra number-theory elementary-number-theory proof-verification recreational-mathematics
asked Jul 26 at 22:45
Red
1,747733
We know that $a^2 + b^2 = c^2$ has infinitely integer solutions which can be written as 3-tuples for example (3,4,5).
Can we conjecture that $$x_1^n +x_2^n + cdots +x_n^n = y^n$$ has infinitely many integer solutions for each $n in mathbb N$? If so how would one prove this to be correct?
I apologize if this is a simple question. I havent taken number theory but this is something that has caught my curiosity. Thanks in advance.
linear-algebra number-theory elementary-number-theory proof-verification recreational-mathematics
asked Jul 26 at 22:45
Red
1,747733
edited Jul 27 at 6:48
asked Jul 26 at 22:45
Red
1,747733
asked Jul 26 at 22:45
Red
1,747733
asked Jul 26 at 22:45
Red
1,747733
1,747733
1
related: mathoverflow.net/questions/62820/pythagorean-5-tuples
– Michael McGovern
Jul 26 at 22:52
1
An answer to this would follow from solutions to Waring's problem.
– Somos
Jul 27 at 0:00
Waring's problem is about the worst case, e.g., the number of 6th powers necessary to express each number as a sum. That number will be greater than 6, so solving Waring won't tell you anything about writing a 6th power as a sum of 6 6th powers.
– Gerry Myerson
Jul 27 at 7:03
1
But, Red, you might want to insist on positive integer solutions, else you can have $x_1=y$, $x_2=cdots=x_n=0$.
– Gerry Myerson
Jul 27 at 7:04
According to en.wikipedia.org/wiki/Sixth_power#Sums "no examples are yet known of a sixth power expressible as the sum of just six sixth powers."
– Gerry Myerson
Jul 27 at 7:07
 |Â
show 1 more comment
1
related: mathoverflow.net/questions/62820/pythagorean-5-tuples
– Michael McGovern
Jul 26 at 22:52
1
An answer to this would follow from solutions to Waring's problem.
– Somos
Jul 27 at 0:00
Waring's problem is about the worst case, e.g., the number of 6th powers necessary to express each number as a sum. That number will be greater than 6, so solving Waring won't tell you anything about writing a 6th power as a sum of 6 6th powers.
– Gerry Myerson
Jul 27 at 7:03
1
But, Red, you might want to insist on positive integer solutions, else you can have $x_1=y$, $x_2=cdots=x_n=0$.
– Gerry Myerson
Jul 27 at 7:04
According to en.wikipedia.org/wiki/Sixth_power#Sums "no examples are yet known of a sixth power expressible as the sum of just six sixth powers."
– Gerry Myerson
Jul 27 at 7:07
1
1
related: mathoverflow.net/questions/62820/pythagorean-5-tuples
– Michael McGovern
Jul 26 at 22:52
related: mathoverflow.net/questions/62820/pythagorean-5-tuples
– Michael McGovern
Jul 26 at 22:52
1
1
An answer to this would follow from solutions to Waring's problem.
– Somos
Jul 27 at 0:00
An answer to this would follow from solutions to Waring's problem.
– Somos
Jul 27 at 0:00
Waring's problem is about the worst case, e.g., the number of 6th powers necessary to express each number as a sum. That number will be greater than 6, so solving Waring won't tell you anything about writing a 6th power as a sum of 6 6th powers.
– Gerry Myerson
Jul 27 at 7:03
Waring's problem is about the worst case, e.g., the number of 6th powers necessary to express each number as a sum. That number will be greater than 6, so solving Waring won't tell you anything about writing a 6th power as a sum of 6 6th powers.
– Gerry Myerson
Jul 27 at 7:03
1
1
But, Red, you might want to insist on positive integer solutions, else you can have $x_1=y$, $x_2=cdots=x_n=0$.
– Gerry Myerson
Jul 27 at 7:04
But, Red, you might want to insist on positive integer solutions, else you can have $x_1=y$, $x_2=cdots=x_n=0$.
– Gerry Myerson
Jul 27 at 7:04
According to en.wikipedia.org/wiki/Sixth_power#Sums "no examples are yet known of a sixth power expressible as the sum of just six sixth powers."
– Gerry Myerson
Jul 27 at 7:07
According to en.wikipedia.org/wiki/Sixth_power#Sums "no examples are yet known of a sixth power expressible as the sum of just six sixth powers."
– Gerry Myerson
Jul 27 at 7:07
 |Â
show 1 more comment
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1
related: mathoverflow.net/questions/62820/pythagorean-5-tuples
– Michael McGovern
Jul 26 at 22:52
1
An answer to this would follow from solutions to Waring's problem.
– Somos
Jul 27 at 0:00
Waring's problem is about the worst case, e.g., the number of 6th powers necessary to express each number as a sum. That number will be greater than 6, so solving Waring won't tell you anything about writing a 6th power as a sum of 6 6th powers.
– Gerry Myerson
Jul 27 at 7:03
1
But, Red, you might want to insist on positive integer solutions, else you can have $x_1=y$, $x_2=cdots=x_n=0$.
– Gerry Myerson
Jul 27 at 7:04
According to en.wikipedia.org/wiki/Sixth_power#Sums "no examples are yet known of a sixth power expressible as the sum of just six sixth powers."
– Gerry Myerson
Jul 27 at 7:07