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 Curious Consecutive Conundrum (Posted on 2008-04-08)
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

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 need a hint | Comment 6 of 7 |
`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=1The 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 equationusing 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 evenand 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 ofgiven pell's equation.I need help from here.KS, can you provide with a hint how to proceed fromhere. I tried induction. It didn't work.`

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

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