Pairs of perfect squares Two perfect squares are said to be friendly if one is obtained from the other by adding
the digit 1 on the left. For example, $1225 = 35 ^ 2$ and $225 = 15 ^ 2$ are friendly. Prove that there are
in finite pairs of friendly and odd perfect squares.
Maybe Any product of square-full numbers is square-full. 
$4n(n+1)+1=4n^{2}+4n+1=(2n+1)^{2}$
I don't know if that would help: http://oeis.org/A060355

Be $a, b$ belonging to integers, we want to show that for all  $a, b$ belonging to integers, there is $ k $ belonging to the natural such that they are infinite.
$a ^ 2 - b ^ 2 = 10 ^ k$
$(a + b) (a-b) = 2^k. 5 ^ k $
$\begin{cases}
a + b = 2 ^ {k-1} \\
 a - b = 2.5 ^ k
\end{cases}$
$2a = 2.5 ^ k + 2 ^ {k-1}$
$a = 5 ^ k + 2 ^ {k-1}$ 
$ 2b = 2 ^ {k-1} - 2.5 ^ k$
$b = 2 ^ {k-2} - 5 ^ k$
Taking $a = 2 ^ {k-2} + 5 ^ k$
$b = 5 ^ k - 2 ^ {k-2}$
Take: $a ^ 2 - b ^ 2 = 2 ^ {2k - 4} + 2.2 ^ {k-2} .5 ^ k + 5 ^ 2k - 5 ^ 2k + 2.2 ^ {k-2} .5 ^ k- 2 ^ {2k-4}$
$a ^ 2-b ^ 2 = 2 ^ k.5 ^ k$
$a ^ 2-b ^ 2 = 10 ^ k$
So we have k belonging to the natural such that $ k \geq 2$. We have infinite numbers
$a = 2 ^ {k-1} + 5 ^ k$
$b = 5 ^ k - 2 ^ {k-2}$ such that $a ^ 2$ and $b ^ 2$ are friendly and a, b are odd
