# Forty-Eight: What is the next bigger number?

The magic word is

Pell equation: https://en.wikipedia.org/wiki/Pell's_equation

What the puzzle actually does is: The smaller square $x^2=25$ equals $\frac{1}{2}48+1$, and the larger square $y^2=49$ equals $48+1$. By multiplying the first equation by $2$ and then subtracting the second from it we get the (negative) Pell equation

$2x^2-y^2=1$

The solutions of this Pell equation are given by the following sequence $(x_n,y_n)$:

$x_1=1$ and $y_1=1$
$x_{n+1}=3x_n+2y_n$ and $y_{n+1}=4x_n+3y_n$

Here $(x_2,y_2)=(5,7)$ corresponds to the given solution, and $(x_3,y_3)=(29,41)$ corresponds to the solution you are looking for. The corresponding magic number and

the answer to the puzzle is: 1680

with $x_3^2=\frac{1}{2}1680+1$ and $y_3^2=1680+1$.

The answer can also be found in the "The On-Line Encyclopedia of Integer Sequences", as sequence A008845. The next five terms (and magic numbers) are $57120$, $1940448$, $65918160$, $2239277040$, and $76069501248$.

The smallest i could get was $1680$. ($1681=41^2$,$841=29^2$).