On relation between prime numbers and exponential Diophantine equation $c\cdot a^x\pm b=z$, feat. $71999999\cdots$

Question

While dealing with some integers which are the elements of the following set $$\{p\mid p\in\mathbb{P}, p=72\times(10^n)-1\}$$

I've could observed that when $n\in\{6,7,8,9\}$, they are all primes.(Such as $71999999$ when $n=6$)

So I've come to wonder the following two questions.

1. It is hard to imagine that there exist a Diophantine equation $a^n\pm b=z$, that is $z\in \mathbb{P} $ for $\{n\mid 1\le n\leq \infty, n\in \mathbb{N}\}$. Can we prove this or would this be a conjecture? - (Edit: Clear, see the comment)
2. As I've shown above, I've dealt with the case $p=72\times(10^n)-1$, which has the sequence of consecutive primes for $n\in\{6,7,8,9\}$. Would there be any exponential Diophantine equation $a^n\pm b=z$ which has longer length of consecutive primes than I've observed? For example, if there is any exponential Diophantine equation $a^n\pm b=z$ that is prime for $n\in\{6,7,8,9,10\}$, then we can call that formula has some longer length of consecutive primes than the one I've observed with. Maybe we can compare them as $length (5) > length (4)$. I'm sure there should be some longer length for some equation than I've got. So if you know any of them, unless it is for some publication, I would like to ask you share the result on here answers so many site users can be shared with those results.

Thanks.

Answer 1

Some comments which are too long to fit in a comment:

Your question is about expressions of the form $a^n+b$, but your example $72 \times 10^n-1$ is not of this form. Rather, it has the form $ca^n+b$ with $c=72$, $a=10$ and $b=-1$. This can make a difference when searching for longer stretches of values of such expressions which are prime.

There's no reason to expect that four consecutive prime values are the global maximum. Indeed, a quick search reveals that $72 \times 10^n-833$ is prime for $n=2,3,4,5,6$. Even better, $7 \times 10^n-927$ is prime for $n=3,4,\ldots,11$. That's $9$ consecutive prime values.

If you really prefer expressions of the form $a^n+b$, without the $c$ coefficient, you can go with $10^n-93$ which is prime for $n=2,3,\ldots,7$, or $2^n+165$ which is prime for $n=3,4,\ldots,9$.