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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    
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Some Thoughts first thoughts | Comment 1 of 7

We have

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

which seems to offer some promise. For the moment though, I prefer pursuing another lead, namely

3L^2 + 3L - P^2 +1 = 0

using the quadratic equation, we see L is sqr(12 P^2 - 3)/6 - 1/2.

Clearly the term inside the square root sign must be a perfect square,

3(4P^2 - 1) = M^2

but the left hand side is divisble by 3, so we can right M = 3Q and substitute R = 2P giving

R^2 - 3Q^2 = 1

This is Pell's equation with fundamental solution R[0]=2, Q[0]=1. All other solutions are known to be given by the sequence

R[N] = 2 R[N-1] + 3 Q[N-1]

Q[N] = R[N-1] + 2 Q[N-1].

Recapituatling results to this point, we see that 2P = R and Q = 2L +1. Notice that elements in the Q sequence are alternately even and odd, parity considerations lead us to conclude that only even N correspond to a solution in P,L.

Having examined P and L with their clothes off, so to speak, thereby inheriting the full knowledge set of the Pell equation, let's dress the back up again as P and L, remembering that only the even terms in the N series are valid:

P[J+1] = 7 P[J] + 12 L[J] + 6

L[J+1] = 4 P[J] + 7 L[J] + 3

It is convenient to combine these equations to define  T[J] = P[J] - sqr(3) L[J]  then

T[J] = (7 - 4 sqr(3)) T[J-1] + 6 - 3 sqr(3)    leading to

2 T[J] = {2 - sqr(3)} {7 - 4 sqr(3)}^J + sqr(3)

we identify P[J] as the sum of all terms not including a multiple of sqr(3). This is easily computed using the binomial expansion:

P[J] = Sum 7^(J-2L)*48^L *J!/[(J-2L)! (2L)!]

        + 6* Sum 7^(J-2L-1)*48^L  *J!/[(J-2L-1)!(2L+1)!]

where both sums from from L=0 up to [J/2]

 

... more to come

Edited on April 10, 2008, 11:34 am
  Posted by FrankM on 2008-04-08 22:58:08

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