Number theory on square numbers [duplicate]












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This question already has an answer here:




  • Number of integer solutions of $x^2 + y^2 = k$

    3 answers




Can a square number be split as the sum of two other squares in two different ways?
The Pythagorean numbers help us to identify numbers which can be split as sum of two squares. Is this unique?










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marked as duplicate by Saad, Jyrki Lahtonen, Ben, Lord Shark the Unknown, apnorton Dec 17 '18 at 7:07


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 3




    $begingroup$
    Number of integer solutions of $x^2 + y^2 = k$ gives all the information you need. And then some. Voting to close as a duplicate.
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    – Jyrki Lahtonen
    Dec 17 '18 at 5:24


















0












$begingroup$



This question already has an answer here:




  • Number of integer solutions of $x^2 + y^2 = k$

    3 answers




Can a square number be split as the sum of two other squares in two different ways?
The Pythagorean numbers help us to identify numbers which can be split as sum of two squares. Is this unique?










share|cite|improve this question









$endgroup$



marked as duplicate by Saad, Jyrki Lahtonen, Ben, Lord Shark the Unknown, apnorton Dec 17 '18 at 7:07


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 3




    $begingroup$
    Number of integer solutions of $x^2 + y^2 = k$ gives all the information you need. And then some. Voting to close as a duplicate.
    $endgroup$
    – Jyrki Lahtonen
    Dec 17 '18 at 5:24
















0












0








0





$begingroup$



This question already has an answer here:




  • Number of integer solutions of $x^2 + y^2 = k$

    3 answers




Can a square number be split as the sum of two other squares in two different ways?
The Pythagorean numbers help us to identify numbers which can be split as sum of two squares. Is this unique?










share|cite|improve this question









$endgroup$





This question already has an answer here:




  • Number of integer solutions of $x^2 + y^2 = k$

    3 answers




Can a square number be split as the sum of two other squares in two different ways?
The Pythagorean numbers help us to identify numbers which can be split as sum of two squares. Is this unique?





This question already has an answer here:




  • Number of integer solutions of $x^2 + y^2 = k$

    3 answers








number-theory






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asked Dec 17 '18 at 1:39









Narayanan RamanNarayanan Raman

1




1




marked as duplicate by Saad, Jyrki Lahtonen, Ben, Lord Shark the Unknown, apnorton Dec 17 '18 at 7:07


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Saad, Jyrki Lahtonen, Ben, Lord Shark the Unknown, apnorton Dec 17 '18 at 7:07


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 3




    $begingroup$
    Number of integer solutions of $x^2 + y^2 = k$ gives all the information you need. And then some. Voting to close as a duplicate.
    $endgroup$
    – Jyrki Lahtonen
    Dec 17 '18 at 5:24
















  • 3




    $begingroup$
    Number of integer solutions of $x^2 + y^2 = k$ gives all the information you need. And then some. Voting to close as a duplicate.
    $endgroup$
    – Jyrki Lahtonen
    Dec 17 '18 at 5:24










3




3




$begingroup$
Number of integer solutions of $x^2 + y^2 = k$ gives all the information you need. And then some. Voting to close as a duplicate.
$endgroup$
– Jyrki Lahtonen
Dec 17 '18 at 5:24






$begingroup$
Number of integer solutions of $x^2 + y^2 = k$ gives all the information you need. And then some. Voting to close as a duplicate.
$endgroup$
– Jyrki Lahtonen
Dec 17 '18 at 5:24












3 Answers
3






active

oldest

votes


















3












$begingroup$

By the Brahmagupta–Fibonacci identity, which states that $$(a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-bc)^2$$ you can get a number which is a sum of two squares two different ways by multiplying two sums of two squares.



So from two Pythagorean triples $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ , the left hand side simplifies to $(a_1a_2)^2$, you can get a number that can be written as a sum of two squares.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Love whipping out the Brahmagupta-Fibonacci identity. math.stackexchange.com/questions/1012733/…
    $endgroup$
    – Matt Samuel
    Dec 17 '18 at 2:40






  • 1




    $begingroup$
    Of course if you have $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ then you also have $(a_2a)^2 + (a_2b)^2 = (a_1a_2)^2$ and $(a_1c)^2 + (a_1d)^2 = (a_1a_2)^2.$ This is how $65^2$ comes to be the sum of two squares in four different ways: take $a=4,b=3,c=12,d=5.$
    $endgroup$
    – David K
    Dec 17 '18 at 12:00



















0












$begingroup$

HINT



$3,4,5$ is a PPT, meaning that any multiple (like $6,8,10$) is a PT



$5,12,13$ is a PPT as well ...






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Why the downvote? Not a good hint?
    $endgroup$
    – Bram28
    Dec 17 '18 at 2:02



















0












$begingroup$

For example
begin{eqnarray*}
65^2=63^2+16^2=33^2+56^2.
end{eqnarray*}






share|cite|improve this answer









$endgroup$













  • $begingroup$
    In fact this happens infinitely often, see this sequence of the Online Encyclopedia of Integer Sequences.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 2:26










  • $begingroup$
    In addition, there are the obvious $65^2=26^2+52^2=25^2+60^2$ (multiples of $3,4,5$ and $5,12,13$). So that's not just two, but four ways for $65^2.$
    $endgroup$
    – David K
    Dec 17 '18 at 2:43












  • $begingroup$
    @BrianHopkins Most of the numbers in that list are not squares, but of course some are.
    $endgroup$
    – David K
    Dec 17 '18 at 2:46






  • 1




    $begingroup$
    @DavidK Oops, yes: of the many related sequences, I think A009177 is more relevant.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 3:15












  • $begingroup$
    Thanks for the solution and illuminating discussion
    $endgroup$
    – Narayanan Raman
    Dec 30 '18 at 1:03


















3 Answers
3






active

oldest

votes








3 Answers
3






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

By the Brahmagupta–Fibonacci identity, which states that $$(a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-bc)^2$$ you can get a number which is a sum of two squares two different ways by multiplying two sums of two squares.



So from two Pythagorean triples $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ , the left hand side simplifies to $(a_1a_2)^2$, you can get a number that can be written as a sum of two squares.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Love whipping out the Brahmagupta-Fibonacci identity. math.stackexchange.com/questions/1012733/…
    $endgroup$
    – Matt Samuel
    Dec 17 '18 at 2:40






  • 1




    $begingroup$
    Of course if you have $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ then you also have $(a_2a)^2 + (a_2b)^2 = (a_1a_2)^2$ and $(a_1c)^2 + (a_1d)^2 = (a_1a_2)^2.$ This is how $65^2$ comes to be the sum of two squares in four different ways: take $a=4,b=3,c=12,d=5.$
    $endgroup$
    – David K
    Dec 17 '18 at 12:00
















3












$begingroup$

By the Brahmagupta–Fibonacci identity, which states that $$(a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-bc)^2$$ you can get a number which is a sum of two squares two different ways by multiplying two sums of two squares.



So from two Pythagorean triples $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ , the left hand side simplifies to $(a_1a_2)^2$, you can get a number that can be written as a sum of two squares.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Love whipping out the Brahmagupta-Fibonacci identity. math.stackexchange.com/questions/1012733/…
    $endgroup$
    – Matt Samuel
    Dec 17 '18 at 2:40






  • 1




    $begingroup$
    Of course if you have $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ then you also have $(a_2a)^2 + (a_2b)^2 = (a_1a_2)^2$ and $(a_1c)^2 + (a_1d)^2 = (a_1a_2)^2.$ This is how $65^2$ comes to be the sum of two squares in four different ways: take $a=4,b=3,c=12,d=5.$
    $endgroup$
    – David K
    Dec 17 '18 at 12:00














3












3








3





$begingroup$

By the Brahmagupta–Fibonacci identity, which states that $$(a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-bc)^2$$ you can get a number which is a sum of two squares two different ways by multiplying two sums of two squares.



So from two Pythagorean triples $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ , the left hand side simplifies to $(a_1a_2)^2$, you can get a number that can be written as a sum of two squares.






share|cite|improve this answer









$endgroup$



By the Brahmagupta–Fibonacci identity, which states that $$(a^2 + b^2)(c^2 + d^2) = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-bc)^2$$ you can get a number which is a sum of two squares two different ways by multiplying two sums of two squares.



So from two Pythagorean triples $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ , the left hand side simplifies to $(a_1a_2)^2$, you can get a number that can be written as a sum of two squares.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 17 '18 at 1:59









twnlytwnly

1,1221213




1,1221213












  • $begingroup$
    Love whipping out the Brahmagupta-Fibonacci identity. math.stackexchange.com/questions/1012733/…
    $endgroup$
    – Matt Samuel
    Dec 17 '18 at 2:40






  • 1




    $begingroup$
    Of course if you have $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ then you also have $(a_2a)^2 + (a_2b)^2 = (a_1a_2)^2$ and $(a_1c)^2 + (a_1d)^2 = (a_1a_2)^2.$ This is how $65^2$ comes to be the sum of two squares in four different ways: take $a=4,b=3,c=12,d=5.$
    $endgroup$
    – David K
    Dec 17 '18 at 12:00


















  • $begingroup$
    Love whipping out the Brahmagupta-Fibonacci identity. math.stackexchange.com/questions/1012733/…
    $endgroup$
    – Matt Samuel
    Dec 17 '18 at 2:40






  • 1




    $begingroup$
    Of course if you have $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ then you also have $(a_2a)^2 + (a_2b)^2 = (a_1a_2)^2$ and $(a_1c)^2 + (a_1d)^2 = (a_1a_2)^2.$ This is how $65^2$ comes to be the sum of two squares in four different ways: take $a=4,b=3,c=12,d=5.$
    $endgroup$
    – David K
    Dec 17 '18 at 12:00
















$begingroup$
Love whipping out the Brahmagupta-Fibonacci identity. math.stackexchange.com/questions/1012733/…
$endgroup$
– Matt Samuel
Dec 17 '18 at 2:40




$begingroup$
Love whipping out the Brahmagupta-Fibonacci identity. math.stackexchange.com/questions/1012733/…
$endgroup$
– Matt Samuel
Dec 17 '18 at 2:40




1




1




$begingroup$
Of course if you have $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ then you also have $(a_2a)^2 + (a_2b)^2 = (a_1a_2)^2$ and $(a_1c)^2 + (a_1d)^2 = (a_1a_2)^2.$ This is how $65^2$ comes to be the sum of two squares in four different ways: take $a=4,b=3,c=12,d=5.$
$endgroup$
– David K
Dec 17 '18 at 12:00




$begingroup$
Of course if you have $a^2 + b^2 = a_1^2$ and $c^2 + d^2 = a_2^2$ then you also have $(a_2a)^2 + (a_2b)^2 = (a_1a_2)^2$ and $(a_1c)^2 + (a_1d)^2 = (a_1a_2)^2.$ This is how $65^2$ comes to be the sum of two squares in four different ways: take $a=4,b=3,c=12,d=5.$
$endgroup$
– David K
Dec 17 '18 at 12:00











0












$begingroup$

HINT



$3,4,5$ is a PPT, meaning that any multiple (like $6,8,10$) is a PT



$5,12,13$ is a PPT as well ...






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Why the downvote? Not a good hint?
    $endgroup$
    – Bram28
    Dec 17 '18 at 2:02
















0












$begingroup$

HINT



$3,4,5$ is a PPT, meaning that any multiple (like $6,8,10$) is a PT



$5,12,13$ is a PPT as well ...






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Why the downvote? Not a good hint?
    $endgroup$
    – Bram28
    Dec 17 '18 at 2:02














0












0








0





$begingroup$

HINT



$3,4,5$ is a PPT, meaning that any multiple (like $6,8,10$) is a PT



$5,12,13$ is a PPT as well ...






share|cite|improve this answer









$endgroup$



HINT



$3,4,5$ is a PPT, meaning that any multiple (like $6,8,10$) is a PT



$5,12,13$ is a PPT as well ...







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 17 '18 at 1:51









Bram28Bram28

63.7k44793




63.7k44793








  • 1




    $begingroup$
    Why the downvote? Not a good hint?
    $endgroup$
    – Bram28
    Dec 17 '18 at 2:02














  • 1




    $begingroup$
    Why the downvote? Not a good hint?
    $endgroup$
    – Bram28
    Dec 17 '18 at 2:02








1




1




$begingroup$
Why the downvote? Not a good hint?
$endgroup$
– Bram28
Dec 17 '18 at 2:02




$begingroup$
Why the downvote? Not a good hint?
$endgroup$
– Bram28
Dec 17 '18 at 2:02











0












$begingroup$

For example
begin{eqnarray*}
65^2=63^2+16^2=33^2+56^2.
end{eqnarray*}






share|cite|improve this answer









$endgroup$













  • $begingroup$
    In fact this happens infinitely often, see this sequence of the Online Encyclopedia of Integer Sequences.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 2:26










  • $begingroup$
    In addition, there are the obvious $65^2=26^2+52^2=25^2+60^2$ (multiples of $3,4,5$ and $5,12,13$). So that's not just two, but four ways for $65^2.$
    $endgroup$
    – David K
    Dec 17 '18 at 2:43












  • $begingroup$
    @BrianHopkins Most of the numbers in that list are not squares, but of course some are.
    $endgroup$
    – David K
    Dec 17 '18 at 2:46






  • 1




    $begingroup$
    @DavidK Oops, yes: of the many related sequences, I think A009177 is more relevant.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 3:15












  • $begingroup$
    Thanks for the solution and illuminating discussion
    $endgroup$
    – Narayanan Raman
    Dec 30 '18 at 1:03
















0












$begingroup$

For example
begin{eqnarray*}
65^2=63^2+16^2=33^2+56^2.
end{eqnarray*}






share|cite|improve this answer









$endgroup$













  • $begingroup$
    In fact this happens infinitely often, see this sequence of the Online Encyclopedia of Integer Sequences.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 2:26










  • $begingroup$
    In addition, there are the obvious $65^2=26^2+52^2=25^2+60^2$ (multiples of $3,4,5$ and $5,12,13$). So that's not just two, but four ways for $65^2.$
    $endgroup$
    – David K
    Dec 17 '18 at 2:43












  • $begingroup$
    @BrianHopkins Most of the numbers in that list are not squares, but of course some are.
    $endgroup$
    – David K
    Dec 17 '18 at 2:46






  • 1




    $begingroup$
    @DavidK Oops, yes: of the many related sequences, I think A009177 is more relevant.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 3:15












  • $begingroup$
    Thanks for the solution and illuminating discussion
    $endgroup$
    – Narayanan Raman
    Dec 30 '18 at 1:03














0












0








0





$begingroup$

For example
begin{eqnarray*}
65^2=63^2+16^2=33^2+56^2.
end{eqnarray*}






share|cite|improve this answer









$endgroup$



For example
begin{eqnarray*}
65^2=63^2+16^2=33^2+56^2.
end{eqnarray*}







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 17 '18 at 1:51









Donald SplutterwitDonald Splutterwit

22.9k21446




22.9k21446












  • $begingroup$
    In fact this happens infinitely often, see this sequence of the Online Encyclopedia of Integer Sequences.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 2:26










  • $begingroup$
    In addition, there are the obvious $65^2=26^2+52^2=25^2+60^2$ (multiples of $3,4,5$ and $5,12,13$). So that's not just two, but four ways for $65^2.$
    $endgroup$
    – David K
    Dec 17 '18 at 2:43












  • $begingroup$
    @BrianHopkins Most of the numbers in that list are not squares, but of course some are.
    $endgroup$
    – David K
    Dec 17 '18 at 2:46






  • 1




    $begingroup$
    @DavidK Oops, yes: of the many related sequences, I think A009177 is more relevant.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 3:15












  • $begingroup$
    Thanks for the solution and illuminating discussion
    $endgroup$
    – Narayanan Raman
    Dec 30 '18 at 1:03


















  • $begingroup$
    In fact this happens infinitely often, see this sequence of the Online Encyclopedia of Integer Sequences.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 2:26










  • $begingroup$
    In addition, there are the obvious $65^2=26^2+52^2=25^2+60^2$ (multiples of $3,4,5$ and $5,12,13$). So that's not just two, but four ways for $65^2.$
    $endgroup$
    – David K
    Dec 17 '18 at 2:43












  • $begingroup$
    @BrianHopkins Most of the numbers in that list are not squares, but of course some are.
    $endgroup$
    – David K
    Dec 17 '18 at 2:46






  • 1




    $begingroup$
    @DavidK Oops, yes: of the many related sequences, I think A009177 is more relevant.
    $endgroup$
    – Brian Hopkins
    Dec 17 '18 at 3:15












  • $begingroup$
    Thanks for the solution and illuminating discussion
    $endgroup$
    – Narayanan Raman
    Dec 30 '18 at 1:03
















$begingroup$
In fact this happens infinitely often, see this sequence of the Online Encyclopedia of Integer Sequences.
$endgroup$
– Brian Hopkins
Dec 17 '18 at 2:26




$begingroup$
In fact this happens infinitely often, see this sequence of the Online Encyclopedia of Integer Sequences.
$endgroup$
– Brian Hopkins
Dec 17 '18 at 2:26












$begingroup$
In addition, there are the obvious $65^2=26^2+52^2=25^2+60^2$ (multiples of $3,4,5$ and $5,12,13$). So that's not just two, but four ways for $65^2.$
$endgroup$
– David K
Dec 17 '18 at 2:43






$begingroup$
In addition, there are the obvious $65^2=26^2+52^2=25^2+60^2$ (multiples of $3,4,5$ and $5,12,13$). So that's not just two, but four ways for $65^2.$
$endgroup$
– David K
Dec 17 '18 at 2:43














$begingroup$
@BrianHopkins Most of the numbers in that list are not squares, but of course some are.
$endgroup$
– David K
Dec 17 '18 at 2:46




$begingroup$
@BrianHopkins Most of the numbers in that list are not squares, but of course some are.
$endgroup$
– David K
Dec 17 '18 at 2:46




1




1




$begingroup$
@DavidK Oops, yes: of the many related sequences, I think A009177 is more relevant.
$endgroup$
– Brian Hopkins
Dec 17 '18 at 3:15






$begingroup$
@DavidK Oops, yes: of the many related sequences, I think A009177 is more relevant.
$endgroup$
– Brian Hopkins
Dec 17 '18 at 3:15














$begingroup$
Thanks for the solution and illuminating discussion
$endgroup$
– Narayanan Raman
Dec 30 '18 at 1:03




$begingroup$
Thanks for the solution and illuminating discussion
$endgroup$
– Narayanan Raman
Dec 30 '18 at 1:03



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