My Math Forum

My Math Forum (http://mymathforum.com/math-forums.php)
-   Algebra (http://mymathforum.com/algebra/)
-   -   diophantine reciprocals: amount of solutions (http://mymathforum.com/algebra/42193-diophantine-reciprocals-amount-solutions.html)

gelatine1 March 18th, 2014 01:22 PM

diophantine reciprocals: amount of solutions
 
Hello,

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 ?

mathbalarka March 19th, 2014 12:45 AM

Re: diophantine reciprocals: amount of solutions
 
It'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).

gelatine1 March 19th, 2014 05:58 AM

Re: diophantine reciprocals: amount of solutions
 
Thanks 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

mathbalarka March 19th, 2014 01:48 PM

Re: diophantine reciprocals: amount of solutions
 
OK, 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!

mathbalarka March 23rd, 2014 01:24 AM

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



All times are GMT -8. The time now is 09:10 PM.

Copyright © 2019 My Math Forum. All rights reserved.