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- - **diophantine reciprocals: amount of solutions**
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diophantine reciprocals: amount of solutionsHello, I was reading about solving an equation of the form on this website: http://www.cut-the-knot.org/arithmet...shtml#solution I understood the derivation of the solutions but now I am wondering how to quickly determine the amount of distinct solutions of the equation with a given z. the solutions are Is there someone who could help me out with this ? |

Re: diophantine reciprocals: amount of solutionsIt's a good exercise for you to prove that there are solutions in total. Now subtract the ones you get modulo the symmetric equivalence (x, y) ~ (y, x). |

Re: diophantine reciprocals: amount of solutionsThanks for your reply. I have no clue how to even start the proof of that. I am not quite familiar with those. But I think it is interesting so maybe you could give me a hint on how to start the proof? Thanks in advance |

Re: diophantine reciprocals: amount of solutionsOK, here's a method : Now let so that this is Which is, by rearranging, We note that , thus divides , i.e., Similarly, divides . Hence the number of solutions can precisely be found by counting Excercise : Do the counting! |

Re: diophantine reciprocals: amount of solutionsA sneaky guy in MSE did some nonobvious trick there which gives essentially a 1-line proof : |

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