About the Pell Equation $x^2 -2y^2=d$, finding $d$ for existence of primitive solutions [on hold]

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The website http://oeis.org/A058529 Shows all Numbers whose prime factors are all congruent to +1 or -1 modulo 8. The first comment says they are exactly the set $D$ of ``numbers of the form $x^2 - 2 y^2$, where x is odd and x and y are relatively prime''. It seems to be correct. However I cannot find out any reference but only some partial results ( If a prime $pin D$ and another $din D$, then $pcdot din D$ ). I hope to know if it is a solved problem or only a conjecture.




elementary-number-theory pythagorean-triples pell-type-equations




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put on hold as off-topic by Dietrich Burde, Ethan Bolker, amWhy, Leucippus, Taroccoesbrocco yesterday


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Dietrich Burde, amWhy, Leucippus, Taroccoesbrocco
If this question can be reworded to fit the rules in the help center, please edit the question.












  • I asked a closely related question, and in fact every prime $p$ of the form $8kpm 1$ has a representation $p=x^2-2y^2$. Since the set $D$ is multiplicatively closed ($Ain D and Bin Dimplies ABin D$) , every number having only prime factors of the form $8kpm 1$ have this representation. Finally, it is easy to show that other prime factors cannot occur. Hence your conjecture is actually true.
    – Peter
    12 hours ago











  • How to show multiplicative closure of $D$ ? I am not so familiar with $mathbb Z[sqrt 2]$, can you give me any hint ? Thank you.
    – user74489
    5 hours ago











  • You do not need that, just consider $$(a^2-2b^2)(c^2-2d^2)=(ac+2bd)^2-2(ad+bc)^2$$ and also consider the parity-check.
    – Peter
    5 hours ago










  • Yes, in fact I can only show the case where $A$ or $B$ is prime.
    – user74489
    5 hours ago










  • I just noticed that this is not the case in general ...
    – Peter
    5 hours ago














up vote
-2
down vote

favorite












The website http://oeis.org/A058529 Shows all Numbers whose prime factors are all congruent to +1 or -1 modulo 8. The first comment says they are exactly the set $D$ of ``numbers of the form $x^2 - 2 y^2$, where x is odd and x and y are relatively prime''. It seems to be correct. However I cannot find out any reference but only some partial results ( If a prime $pin D$ and another $din D$, then $pcdot din D$ ). I hope to know if it is a solved problem or only a conjecture.




elementary-number-theory pythagorean-triples pell-type-equations




share|cite|improve this question











put on hold as off-topic by Dietrich Burde, Ethan Bolker, amWhy, Leucippus, Taroccoesbrocco yesterday


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Dietrich Burde, amWhy, Leucippus, Taroccoesbrocco
If this question can be reworded to fit the rules in the help center, please edit the question.












  • I asked a closely related question, and in fact every prime $p$ of the form $8kpm 1$ has a representation $p=x^2-2y^2$. Since the set $D$ is multiplicatively closed ($Ain D and Bin Dimplies ABin D$) , every number having only prime factors of the form $8kpm 1$ have this representation. Finally, it is easy to show that other prime factors cannot occur. Hence your conjecture is actually true.
    – Peter
    12 hours ago











  • How to show multiplicative closure of $D$ ? I am not so familiar with $mathbb Z[sqrt 2]$, can you give me any hint ? Thank you.
    – user74489
    5 hours ago











  • You do not need that, just consider $$(a^2-2b^2)(c^2-2d^2)=(ac+2bd)^2-2(ad+bc)^2$$ and also consider the parity-check.
    – Peter
    5 hours ago










  • Yes, in fact I can only show the case where $A$ or $B$ is prime.
    – user74489
    5 hours ago










  • I just noticed that this is not the case in general ...
    – Peter
    5 hours ago












up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











The website http://oeis.org/A058529 Shows all Numbers whose prime factors are all congruent to +1 or -1 modulo 8. The first comment says they are exactly the set $D$ of ``numbers of the form $x^2 - 2 y^2$, where x is odd and x and y are relatively prime''. It seems to be correct. However I cannot find out any reference but only some partial results ( If a prime $pin D$ and another $din D$, then $pcdot din D$ ). I hope to know if it is a solved problem or only a conjecture.




elementary-number-theory pythagorean-triples pell-type-equations




share|cite|improve this question











The website http://oeis.org/A058529 Shows all Numbers whose prime factors are all congruent to +1 or -1 modulo 8. The first comment says they are exactly the set $D$ of ``numbers of the form $x^2 - 2 y^2$, where x is odd and x and y are relatively prime''. It seems to be correct. However I cannot find out any reference but only some partial results ( If a prime $pin D$ and another $din D$, then $pcdot din D$ ). I hope to know if it is a solved problem or only a conjecture.





elementary-number-theory pythagorean-triples pell-type-equations





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked 2 days ago









user74489

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put on hold as off-topic by Dietrich Burde, Ethan Bolker, amWhy, Leucippus, Taroccoesbrocco yesterday


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Dietrich Burde, amWhy, Leucippus, Taroccoesbrocco
If this question can be reworded to fit the rules in the help center, please edit the question.




put on hold as off-topic by Dietrich Burde, Ethan Bolker, amWhy, Leucippus, Taroccoesbrocco yesterday


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Dietrich Burde, amWhy, Leucippus, Taroccoesbrocco
If this question can be reworded to fit the rules in the help center, please edit the question.











  • I asked a closely related question, and in fact every prime $p$ of the form $8kpm 1$ has a representation $p=x^2-2y^2$. Since the set $D$ is multiplicatively closed ($Ain D and Bin Dimplies ABin D$) , every number having only prime factors of the form $8kpm 1$ have this representation. Finally, it is easy to show that other prime factors cannot occur. Hence your conjecture is actually true.
    – Peter
    12 hours ago











  • How to show multiplicative closure of $D$ ? I am not so familiar with $mathbb Z[sqrt 2]$, can you give me any hint ? Thank you.
    – user74489
    5 hours ago











  • You do not need that, just consider $$(a^2-2b^2)(c^2-2d^2)=(ac+2bd)^2-2(ad+bc)^2$$ and also consider the parity-check.
    – Peter
    5 hours ago










  • Yes, in fact I can only show the case where $A$ or $B$ is prime.
    – user74489
    5 hours ago










  • I just noticed that this is not the case in general ...
    – Peter
    5 hours ago
















  • I asked a closely related question, and in fact every prime $p$ of the form $8kpm 1$ has a representation $p=x^2-2y^2$. Since the set $D$ is multiplicatively closed ($Ain D and Bin Dimplies ABin D$) , every number having only prime factors of the form $8kpm 1$ have this representation. Finally, it is easy to show that other prime factors cannot occur. Hence your conjecture is actually true.
    – Peter
    12 hours ago











  • How to show multiplicative closure of $D$ ? I am not so familiar with $mathbb Z[sqrt 2]$, can you give me any hint ? Thank you.
    – user74489
    5 hours ago











  • You do not need that, just consider $$(a^2-2b^2)(c^2-2d^2)=(ac+2bd)^2-2(ad+bc)^2$$ and also consider the parity-check.
    – Peter
    5 hours ago










  • Yes, in fact I can only show the case where $A$ or $B$ is prime.
    – user74489
    5 hours ago










  • I just noticed that this is not the case in general ...
    – Peter
    5 hours ago















I asked a closely related question, and in fact every prime $p$ of the form $8kpm 1$ has a representation $p=x^2-2y^2$. Since the set $D$ is multiplicatively closed ($Ain D and Bin Dimplies ABin D$) , every number having only prime factors of the form $8kpm 1$ have this representation. Finally, it is easy to show that other prime factors cannot occur. Hence your conjecture is actually true.
– Peter
12 hours ago





I asked a closely related question, and in fact every prime $p$ of the form $8kpm 1$ has a representation $p=x^2-2y^2$. Since the set $D$ is multiplicatively closed ($Ain D and Bin Dimplies ABin D$) , every number having only prime factors of the form $8kpm 1$ have this representation. Finally, it is easy to show that other prime factors cannot occur. Hence your conjecture is actually true.
– Peter
12 hours ago













How to show multiplicative closure of $D$ ? I am not so familiar with $mathbb Z[sqrt 2]$, can you give me any hint ? Thank you.
– user74489
5 hours ago





How to show multiplicative closure of $D$ ? I am not so familiar with $mathbb Z[sqrt 2]$, can you give me any hint ? Thank you.
– user74489
5 hours ago













You do not need that, just consider $$(a^2-2b^2)(c^2-2d^2)=(ac+2bd)^2-2(ad+bc)^2$$ and also consider the parity-check.
– Peter
5 hours ago




You do not need that, just consider $$(a^2-2b^2)(c^2-2d^2)=(ac+2bd)^2-2(ad+bc)^2$$ and also consider the parity-check.
– Peter
5 hours ago












Yes, in fact I can only show the case where $A$ or $B$ is prime.
– user74489
5 hours ago




Yes, in fact I can only show the case where $A$ or $B$ is prime.
– user74489
5 hours ago












I just noticed that this is not the case in general ...
– Peter
5 hours ago




I just noticed that this is not the case in general ...
– Peter
5 hours ago















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