$n$ or $2n$ is a sum of three squares











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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







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  • 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















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







share|cite|improve this question




















  • 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













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







share|cite|improve this question
















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




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edited Nov 22 at 9:21

























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














  • 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










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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.






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    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.






    share|cite|improve this answer



























      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.






      share|cite|improve this answer

























        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.






        share|cite|improve this answer














        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.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 22 at 20:08

























        answered Oct 15 at 17:38









        Alex R.

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        24.7k12352






























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