My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
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.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 ?
gelatine1 is offline  
 
March 19th, 2014, 12:45 AM   #2
Math Team
 
mathbalarka's Avatar
 
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).
mathbalarka is offline  
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
gelatine1 is offline  
March 19th, 2014, 01:48 PM   #4
Math Team
 
mathbalarka's Avatar
 
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!
mathbalarka is offline  
March 23rd, 2014, 01:24 AM   #5
Math Team
 
mathbalarka's Avatar
 
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 1-line proof :

mathbalarka is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

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





Copyright © 2019 My Math Forum. All rights reserved.