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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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re: What value for L? | Comment 4 of 7 |
(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
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