Numbers of the form $n^k-1$

Question

I know that numbers of the form $2^k-1$ are called Mersenne numbers. But are there other special numbers which are one less than a power of an integer (for instance, does $3^k-1$ have some special significance)? If so, what are they called?

Answer 1

The reason the Mersenne numbers are called in a special way is inherited from the time when mathematicians were looking for generic formulas giving prime numbers.

So, there are the Mersenne numbers, but also the Fermat numbers. Fermat thought that this formula would give only primes, until Euler proved that $$ 641\mid 4294967297 $$ No one believes that every Fermat number except the 4 first are not prime.

Back to the powers $-1$, one can easily prove that $$ a-b\mid a^n - b^n $$

so if $a^n - 1$ is prime with $n>1$, then $a-1 = 1$. Which is why Mersenne numbers have a name and not the numbers in the form $a^n-1$ for any other $a$.

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NB: this lemma also allows a quick proof of $2^p - 1$ is prime $\implies p$ is prime.