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Solve in integers (Posted on 2014-12-17) Difficulty: 2 of 5
Find all integer solutions of:
x*(x+y)=y2+1

No Solution Yet Submitted by Ady TZIDON    
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Solution answer | Comment 1 of 3
The values of x are the generalized Pell numbers P(n,5,5)
(OEIS A141448) beginning with 1: (1, 2, 5, 13, 34,...) and their associative negative equivalent (-1, -2, -5, -13, -34,...).

The values of y are the bisection of the Fibonacci sequence F(2n) (OEIS A001906):  (0, 1, 3, 8, 21, 55,...) and their associative negative equivalent (0, -1, -3, -8, -21, -55, ....).

There exists a two-to-one correspondence between x and y, where each value maps to two values of the other. This can be observed by looking at the set of numbers (x,y):

Below I present the pairs (x, y) as (x, (y1, y2)) where
(x, (y1, y2) = (x,y1) and (x, y2).  I assigned y1 as the "negative" values of y, and y2 as the "positive" values of y. The "positive" values of y are "offset one position upwards" respective to the "negative" y values).  The values of x, as does the corresponding Pell number P(n,5,5) sequence, continues toward infinity, but in this case also to negative infinity. y1 continues toward negative infinity and y2 continues toward infinity as does its corresponding sequence of the bisection of Fibonacci numbers. Thus there are an infinite number of integer solutions.

 ...
(-34, (-55, 21)),
(-13, (-21,  8)),
( -5, ( -8,  3)),
( -2, ( -3,  1)),
( -1, ( -1,  0)),
(  1, (  0,  1)),
(  2, ( -1,  3)),
(  5, ( -3,  8)),
( 13, ( -8, 21)),
( 34, (-21, 55)),
 ...

Edited on December 18, 2014, 11:21 am
  Posted by Dej Mar on 2014-12-17 10:26:40

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