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 What value for L? by brianjn)

Rather than a spreadsheet, I turned to UBASIC.  This little program brute force checks each L from 1 to 100,000,000:

    1   Lmax=10^8
    5   print " L"," P"," X"," Check"
   10   for L=1 to Lmax
   20   S=3*L*(L+1)+1
   30   P=isqrt(S)
   40   if P*P<>S then 100
   50   X=isqrt(P/2)
   60   print L,P,X,2*X*(X+1)+1
  100   next L

The results yielded:

       L         P     X
       7        13     2
     104       181     9
    1455      2521    35
   20272     35113   132
  282359    489061   494
 3932760   6811741  1845
54776287  94875313  6887

The two constraints the program looks for are (L+1)^3 – L^3 = P^2 and X^2 + (X+1)^2 = P.  Just by looking at the output, I would expect the next value to occur with L around 800,000,000.

Posted by Brian Smith on 2008-04-10 00:15:01