**Definition.** A prime number $p$ is called *strange* if there exists $k>1$ such that each prime divisior of $p^k-1$ divides $p-1$.


**Example 3.** The prime number $p=3$ is strange as $3^2-1=8$ has the same prime divisiors as $3-1=2$.


**Example 5.** The prime number $p=5$ is not strange, since for every $k>1$ the number $5^k-1$ is not a power of $2$ (by the [Mihailescu Theorem](https://mathworld.wolfram.com/CatalansConjecture.html)). By the same reason the prime number $p=17$ is not strange.


**Example 7.** The prime number $p=7$ is strange since $7^2-1=48$ has the same prime divisors as $7-1=6$.


**Example 31.** The prime number $p=31$ is strange because $31^2-1=2^6\times 3\times 5$ has the prime divisors as $31-1=2\times 3\times 5$.


Using the small Fermat Theorem, it is possible to prove the following characterization



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> **Theorem.** A prime number $p$ is not strange if and only if for every prime divisor $q$ of $p-1$ the number $p^q-1$ has a prime divisor that does not divide $p-1$.
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This theorem implies that the prime numbers $11,13,19,23,29,37,41,43,47,53,61,67,71,73,79,83,89,97$ are not strange.


Therefore, among prime numbers $<100$ only 3,7, 31 are strange. All these numbers are Mersenne numbers. In his comment Yaakov Baruch observed that each Mersenne number is strange. So, we can ask



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> **Question 1.** Is each strange prime number Mersenne prime?
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> **Question 2.** Is the set of non-strange numbers infinite?
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> **Question 3.** Is it true that for any number $x$ and prime numbers $p\_1,\dots,p\_n$ that not divide $x$, the arithmetic progression $x+p\_1\dots p\_n\mathbb Z$ contains a non-strange prime number?
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Strange and non-strange prime numbers, are there infinitely many of them?
[Zsigmondy's Theorem](https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem) shows that the only strange primes are the Mersenne primes. Indeed, it shows that for any $n\geq 2$ the number $p^n-1$ has a prime factor not dividing $p^k-1$ for any $k<n$, including $k=1$, with the only exceptions being $n=2$ when $p+1$ is a power of two.


It follows there are infinitely many non-strange primes, and existence of infinitely many strange ones is a famous open problem.


