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

Home > Just Math
Curious Consecutive Conundrum (Posted on 2008-04-08) Difficulty: 4 of 5
L and P are positive integers that satisfy this equation:

(L+1)3 L3 = P2

For example, 83 - 73 = 132; 1053 - 1043 = 1812, and so on.

Prove that P is always expressible as the sum of squares of two consecutive positive integers.

(For example, 13 = 22 + 32; 181 = 92 + 102, and so on.)

See The Solution Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
need a hint | Comment 6 of 7 |
3L+3L+1=P
=> 3/4(4L+4L+1)+1/4=P
=> (2P)-3(2L+1)=1 -- (1)
Let x=2P and y=2L+1
=> x-3y=1
The above equation is an example of Pell's equation.
The basic solution is (2,1). Let this be (x(1),y(1)).
There are infinite solutions for the above equation.
The recursive equation for solutions of Pell's equation
using basic solution.
x(i+1)=2x(i)+3y(i) -- (2)
y(i+1)=x(i)+2y(i) -- (3)
(x(2k+1),y(2k+1)) are the solution from eq(1) as x is even
and y must be odd.
Or
x(n)+y(n)√3 = (2+√3)^n -- (4)
In the above equation x(n),y(n) are nth solutions of
given pell's equation.
I need help from here.
KS, can you provide with a hint how to proceed from
here. I tried induction. It didn't work.

  Posted by Praneeth on 2008-07-23 03:40:32
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 (2)
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