Find integers $x$ and $y$ such that $8^x-9^y=431$ Find integers $x$ and $y$ such that $8^x-9^y=431$

My working:
By taking mod 9 and 16, I got $x$ odd and $y$ even.
Also $8^x&gt;431\implies x\ge 3$
For $x=3$ I got $y=2$
We suspect that $512-81$ is the largest solution. Proof by contradiction:

Giving new names to $x,y,$  we say
$$  512(8^x - 1) = 81 (9^y - 1)  $$
We ASSUME both $x \geq 1, y \geq 1.$

First, $8^x \equiv 1 \pmod {81}.$ A calculation shows that $x$ must be divisible by $18$

Next,
$$ 8^{18} - 1 = 2^{54 - 1} =  3^4 \cdot 7 \cdot 19 \cdot 73 \cdot 87211 \cdot  262657 $$

The final prime factor is $262657,$ and this must divide $9^y - 1,$ or
$$ 9^y \equiv 1 \pmod{262657} $$

Yet another calculation (keyword order) tells us that the $y$ is divisible by $$  2^7 \cdot 3 \cdot 19.$$  Then $9^y - 1$ is divisible by $9^{128} - 1,$ and
$$  9^{128} - 1 = 3^{256} - 1 = 2^{10} \cdot 5 \cdot 17 \cdot 41 \cdot 193 \cdot 257 \cdot 275201  \cdot \mbox{BIG}  $$

That's all we needed. We find $9^y-1$ divisible by $1024.$ Therefore $512(8^x - 1) $ is divisible by $1024,$ which is a contradiction of $x \geq 1.$

=============================

The large prime factors of $3^{256} - 1$ are shown in:

? factor( 3^2 + 1)
%3 = 
[2 1]

[5 1]

? factor( 3^4 + 1)
%4 = 
[ 2 1]

[41 1]

? factor( 3^8 + 1)
%5 = 
[  2 1]

[ 17 1]

[193 1]

? factor( 3^16 + 1)
%6 = 
[       2 1]

[21523361 1]

? factor( 3^32 + 1)
%7 = 
[              2 1]

[926510094425921 1]

? factor( 3^64 + 1)
%8 = 
[                              2 1]

[1716841910146256242328924544641 1]

? factor( 3^128 + 1)
%9 = 
[                    2 1]

[                  257 1]

[               275201 1]

[      138424618868737 1]

[  3913786281514524929 1]

[153849834853910661121 1]

? 


==========================
$$ 8^{y+d} - 9^y = 431 $$

$$ 8^d = \frac{431 + 9^y}{8^y} $$

$$ 8^d = (\frac{9}{8})^y+\frac{431}{8^y} $$

$$ 8^d \sim (\frac{9}{8})^y $$

$$ 3d\cdot ln(2) \sim y \cdot(2ln(3)-3ln(2)) $$

$$ d \sim y \cdot(\frac{2\cdot ln(3)-3\cdot ln(2)}{3\cdot ln(2)}) $$

$$ d \sim y \cdot 0.0566416671474374543024926292985... $$

$$ \frac{3}{53} = 0,056603773... $$

First better aproximation, for $d=3$ and $y=53$:

$$ 8^{56} -9^{53} = -1,56579...\cdot10^{48} $$

Etc...

This problem is related to the approximation of $log_2 (3)$ to a rational number.

And as the different rational ones approach.