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.)

(In reply to need a hint by Praneeth)

Move to:    

(2P)²- 1 = 3(2L+1)²     (from (2P)²-3(2L+1)² = 1)

Then, factorize the lhs, and ...... ......

 

Edited on July 23, 2008, 4:19 pm

Posted by K Sengupta on 2008-07-23 16:11:47