$n$ or $2n$ is a sum of three squares
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0
down vote
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For a given positive integer $n$, show that $n$ or $2n$ is a sum of
three squares.
My attempt: If $n$ is a sum of three squares, there is nothing to prove. So assume $n$ is not a sum of three squares. Then $$nequiv7pmod8implies 2nequiv6pmod8$$. I'm completely stuck here. I know that no positive integer of the form $4^a(8b+7)$ can be written as sum of three squares. My textbook (Elementary Number Theory- David Burton)doesn't give the proof of the converse of the statement,but mentions that the converse is true. The question is from the same book. So are we supposed to use the full theorem?(Alex has already given a proof using the converse.) Is there any way to proceed without using the converse of the theorem? Thanks.
number-theory elementary-number-theory
asked Oct 15 at 17:25
Thomas Shelby
1,170116
add a comment |
up vote
0
down vote
favorite
For a given positive integer $n$, show that $n$ or $2n$ is a sum of
three squares.
My attempt: If $n$ is a sum of three squares, there is nothing to prove. So assume $n$ is not a sum of three squares. Then $$nequiv7pmod8implies 2nequiv6pmod8$$. I'm completely stuck here. I know that no positive integer of the form $4^a(8b+7)$ can be written as sum of three squares. My textbook (Elementary Number Theory- David Burton)doesn't give the proof of the converse of the statement,but mentions that the converse is true. The question is from the same book. So are we supposed to use the full theorem?(Alex has already given a proof using the converse.) Is there any way to proceed without using the converse of the theorem? Thanks.
number-theory elementary-number-theory
asked Oct 15 at 17:25
Thomas Shelby
1,170116
1
Could you please recall exactly what you know regarding sum of three squares. It seems you might know Legendre's three-squares theorem, but you do not recall it correctly.
– quid♦
Oct 15 at 17:29
In general, it is a good practice to provide a little context for the questions you post: this helps other members help you by giving perhaps the answer you are searching for.
– Daniele Tampieri
Oct 15 at 17:37
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For a given positive integer $n$, show that $n$ or $2n$ is a sum of
three squares.
My attempt: If $n$ is a sum of three squares, there is nothing to prove. So assume $n$ is not a sum of three squares. Then $$nequiv7pmod8implies 2nequiv6pmod8$$. I'm completely stuck here. I know that no positive integer of the form $4^a(8b+7)$ can be written as sum of three squares. My textbook (Elementary Number Theory- David Burton)doesn't give the proof of the converse of the statement,but mentions that the converse is true. The question is from the same book. So are we supposed to use the full theorem?(Alex has already given a proof using the converse.) Is there any way to proceed without using the converse of the theorem? Thanks.
number-theory elementary-number-theory
asked Oct 15 at 17:25
Thomas Shelby
1,170116
For a given positive integer $n$, show that $n$ or $2n$ is a sum of
three squares.
My attempt: If $n$ is a sum of three squares, there is nothing to prove. So assume $n$ is not a sum of three squares. Then $$nequiv7pmod8implies 2nequiv6pmod8$$. I'm completely stuck here. I know that no positive integer of the form $4^a(8b+7)$ can be written as sum of three squares. My textbook (Elementary Number Theory- David Burton)doesn't give the proof of the converse of the statement,but mentions that the converse is true. The question is from the same book. So are we supposed to use the full theorem?(Alex has already given a proof using the converse.) Is there any way to proceed without using the converse of the theorem? Thanks.
number-theory elementary-number-theory
number-theory elementary-number-theory
asked Oct 15 at 17:25
Thomas Shelby
1,170116
asked Oct 15 at 17:25
Thomas Shelby
1,170116
edited Nov 22 at 9:21
asked Oct 15 at 17:25
Thomas Shelby
1,170116
asked Oct 15 at 17:25
Thomas Shelby
1,170116
asked Oct 15 at 17:25
Thomas Shelby
1,170116
1,170116
1
Could you please recall exactly what you know regarding sum of three squares. It seems you might know Legendre's three-squares theorem, but you do not recall it correctly.
– quid♦
Oct 15 at 17:29
In general, it is a good practice to provide a little context for the questions you post: this helps other members help you by giving perhaps the answer you are searching for.
– Daniele Tampieri
Oct 15 at 17:37
add a comment |
1
Could you please recall exactly what you know regarding sum of three squares. It seems you might know Legendre's three-squares theorem, but you do not recall it correctly.
– quid♦
Oct 15 at 17:29
In general, it is a good practice to provide a little context for the questions you post: this helps other members help you by giving perhaps the answer you are searching for.
– Daniele Tampieri
Oct 15 at 17:37
1
1
Could you please recall exactly what you know regarding sum of three squares. It seems you might know Legendre's three-squares theorem, but you do not recall it correctly.
– quid♦
Oct 15 at 17:29
Could you please recall exactly what you know regarding sum of three squares. It seems you might know Legendre's three-squares theorem, but you do not recall it correctly.
– quid♦
Oct 15 at 17:29
In general, it is a good practice to provide a little context for the questions you post: this helps other members help you by giving perhaps the answer you are searching for.
– Daniele Tampieri
Oct 15 at 17:37
In general, it is a good practice to provide a little context for the questions you post: this helps other members help you by giving perhaps the answer you are searching for.
– Daniele Tampieri
Oct 15 at 17:37
add a comment |
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
So $x$ is the sum of three squares iff $x$ is not of the form $4^a(8b+7)$ via Legendre's theorem. Suppose $n$ is not the sum of three squares, so that $x=4^a(8b+7)$ for some integers $a,b$. Then $2x=2cdot 4^a(8b+7)$. Notice that $8b+7$ is always odd. So it's not possible that $2x=4^c(8d+7)$, as $c=a$ necessarily, e.g. $2cdot 4^a=4^c$ is impossible.
answered Oct 15 at 17:38
Alex R.
24.7k12352
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
So $x$ is the sum of three squares iff $x$ is not of the form $4^a(8b+7)$ via Legendre's theorem. Suppose $n$ is not the sum of three squares, so that $x=4^a(8b+7)$ for some integers $a,b$. Then $2x=2cdot 4^a(8b+7)$. Notice that $8b+7$ is always odd. So it's not possible that $2x=4^c(8d+7)$, as $c=a$ necessarily, e.g. $2cdot 4^a=4^c$ is impossible.
answered Oct 15 at 17:38
Alex R.
24.7k12352
add a comment |
up vote
3
down vote
accepted
So $x$ is the sum of three squares iff $x$ is not of the form $4^a(8b+7)$ via Legendre's theorem. Suppose $n$ is not the sum of three squares, so that $x=4^a(8b+7)$ for some integers $a,b$. Then $2x=2cdot 4^a(8b+7)$. Notice that $8b+7$ is always odd. So it's not possible that $2x=4^c(8d+7)$, as $c=a$ necessarily, e.g. $2cdot 4^a=4^c$ is impossible.
answered Oct 15 at 17:38
Alex R.
24.7k12352
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
So $x$ is the sum of three squares iff $x$ is not of the form $4^a(8b+7)$ via Legendre's theorem. Suppose $n$ is not the sum of three squares, so that $x=4^a(8b+7)$ for some integers $a,b$. Then $2x=2cdot 4^a(8b+7)$. Notice that $8b+7$ is always odd. So it's not possible that $2x=4^c(8d+7)$, as $c=a$ necessarily, e.g. $2cdot 4^a=4^c$ is impossible.
answered Oct 15 at 17:38
Alex R.
24.7k12352
So $x$ is the sum of three squares iff $x$ is not of the form $4^a(8b+7)$ via Legendre's theorem. Suppose $n$ is not the sum of three squares, so that $x=4^a(8b+7)$ for some integers $a,b$. Then $2x=2cdot 4^a(8b+7)$. Notice that $8b+7$ is always odd. So it's not possible that $2x=4^c(8d+7)$, as $c=a$ necessarily, e.g. $2cdot 4^a=4^c$ is impossible.
answered Oct 15 at 17:38
Alex R.
24.7k12352
edited Nov 22 at 20:08
answered Oct 15 at 17:38
Alex R.
24.7k12352
answered Oct 15 at 17:38
Alex R.
24.7k12352
answered Oct 15 at 17:38
Alex R.
24.7k12352
24.7k12352
add a comment |
add a comment |
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Could you please recall exactly what you know regarding sum of three squares. It seems you might know Legendre's three-squares theorem, but you do not recall it correctly.
– quid♦
Oct 15 at 17:29
In general, it is a good practice to provide a little context for the questions you post: this helps other members help you by giving perhaps the answer you are searching for.
– Daniele Tampieri
Oct 15 at 17:37