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.)

3L²+3L+1=P²
=> 3/4(4L²+4L+1)+1/4=P²
=> (2P)²-3(2L+1)²=1 -- (1)
Let x=2P and y=2L+1
=> x²-3y²=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