
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 18th, 2014, 01:22 PM  #1 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11  diophantine reciprocals: amount of solutions
Hello, I was reading about solving an equation of the form on this website: http://www.cuttheknot.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 ? 
March 19th, 2014, 12:45 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  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).

March 19th, 2014, 05:58 AM  #3 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11  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 
March 19th, 2014, 01:48 PM  #4 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  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! 
March 23rd, 2014, 01:24 AM  #5 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: diophantine reciprocals: amount of solutions
A sneaky guy in MSE did some nonobvious trick there which gives essentially a 1line proof : 

Tags 
amount, diophantine, reciprocals, solutions 
Search tags for this page 
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Amount  Chikis  Elementary Math  11  October 22nd, 2013 04:55 PM 
Practical Algebra: Reciprocals  njc  Algebra  3  August 1st, 2013 06:22 AM 
Reciprocals  julieta  Algebra  1  March 18th, 2011 04:31 AM 
Sum of reciprocals of distinct squares of integers  zolden  Number Theory  0  December 14th, 2008 01:32 PM 
Infinite sum of the reciprocals of the Fibonacci numbers.  Infinity  Number Theory  13  July 21st, 2007 08:35 PM 