All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Perfect Square Not (Posted on 2008-05-07) Difficulty: 2 of 5
The pairs of positive integers (A, B) are such that B divides 2A2.

Prove that A2 + B cannot be a perfect square.

See The Solution Submitted by K Sengupta    
Rating: 2.0000 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Another solution | Comment 5 of 6 |

assume A+B=N
Since A<N, we set A+M=N, squaring gives A+2AM+M=N
thus B=2AM+M, and since B | 2A, we can set BK=2A, ie
2AMK+MK=2A.   (1)
Thus 2 | MK. Now assume that K isnt divisible by 2. Then 2 | M, and we can set M=2m, which gives.
4AmK+4mK=2A
2AmK+2mK=A
now we see that 2 | A, and we set A=2a, which gives
4amK+2mK=4a
2amK+mK=2a
but this is the same form as equation (1), thus this process will continue forever, and we have a contradiction. Thus 2 | K, and in (1) we set K=2k, which yields:

4AMk+2Mk=2A
Mk=A-2AMk
adding Mk to both sides:
Mk+Mk=A-2AMk+Mk
Mk(k+1)=(A-Mk)

thus k(k+1) must be a square, but the product of two consecutive positive integers cant be a square.

Edited on May 24, 2008, 3:53 pm
  Posted by Jonathan Lindgren on 2008-05-24 15:52:32

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 (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information