 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Multiply Together, Get Cubic (Posted on 2007-06-21) Determine all possible integer pairs (p,q) such that p+q˛+sł=pqs, where s=gcd(p,q) and gcd denotes the greatest common divisor.

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (1 votes) Comments: ( Back to comment list | You must be logged in to post comments.) if s = 1 (one more solution!) | Comment 4 of 8 | In all cases, p = (q^2 + s^3)/(qs - 1)

If p and q are relatively prime (s = 1), then

p = (q^2 + 1)/(q - 1) = (q - 1) + 2q/(q - 1)

P is therefore integral if q- 1 divides 2 (i.e, q = 3)
or if q - 1 divides q (i.e, q = 2 or 0)

This leads to three solutions:
(5, 3), (5,2) and (-1,0)

I checked the definitiion, and the gcd of -1 and 0 is in fact 1!

Next step: s > 1

Edited on June 21, 2007, 7:01 pm
 Posted by Steve Herman on 2007-06-21 18:59:58 Please log in:
 Login: Password: Remember me: Sign up! | Forgot password

 Search: Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2021 by Animus Pactum Consulting. All rights reserved. Privacy Information