How do I prove that $mathcal{P}_d(mathbb{Z})^+neqvarnothing$ and that Pell's equation solutions are...












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I recently had to skip a number theory lecture because I was sick, and they proved that the set



$$mathcal{P}_d(mathbb{Z})^+=lbrace(x_0,y_0)inmathbb{Z}^2_{geq1}:x_0^2-dy_0^2=1rbraceneqvarnothing$$
And that if $(a,b)inmathcal{P}_d(mathbb{Z})^+$ then $frac ab$ is a convergent for $sqrt d$.



I tried proving those two statements but I just can't seem to get even near a proof, could someone cite a paper or show me a proof for both statements?






number-theory continued-fractions pell-type-equations







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    1












    $begingroup$


    I recently had to skip a number theory lecture because I was sick, and they proved that the set



    $$mathcal{P}_d(mathbb{Z})^+=lbrace(x_0,y_0)inmathbb{Z}^2_{geq1}:x_0^2-dy_0^2=1rbraceneqvarnothing$$
    And that if $(a,b)inmathcal{P}_d(mathbb{Z})^+$ then $frac ab$ is a convergent for $sqrt d$.



    I tried proving those two statements but I just can't seem to get even near a proof, could someone cite a paper or show me a proof for both statements?






    number-theory continued-fractions pell-type-equations







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I recently had to skip a number theory lecture because I was sick, and they proved that the set



      $$mathcal{P}_d(mathbb{Z})^+=lbrace(x_0,y_0)inmathbb{Z}^2_{geq1}:x_0^2-dy_0^2=1rbraceneqvarnothing$$
      And that if $(a,b)inmathcal{P}_d(mathbb{Z})^+$ then $frac ab$ is a convergent for $sqrt d$.



      I tried proving those two statements but I just can't seem to get even near a proof, could someone cite a paper or show me a proof for both statements?






      number-theory continued-fractions pell-type-equations







      share|cite|improve this question











      $endgroup$




      I recently had to skip a number theory lecture because I was sick, and they proved that the set



      $$mathcal{P}_d(mathbb{Z})^+=lbrace(x_0,y_0)inmathbb{Z}^2_{geq1}:x_0^2-dy_0^2=1rbraceneqvarnothing$$
      And that if $(a,b)inmathcal{P}_d(mathbb{Z})^+$ then $frac ab$ is a convergent for $sqrt d$.



      I tried proving those two statements but I just can't seem to get even near a proof, could someone cite a paper or show me a proof for both statements?






      number-theory continued-fractions pell-type-equations




      number-theory continued-fractions pell-type-equations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 17 '18 at 12:57







      Bruno Andrades

















      asked Dec 16 '18 at 23:48









      Bruno AndradesBruno Andrades

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