Find all integer solutions of:
x*(x+y)=y²+1

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