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I saw on this website that $7$ is the only prime number followed by $x^3$. What is known about primes followed by $x^n$, for $n>3$? Are there any?

For less than $n=2$, there is clearly $3$ which is followed by $2^2$. That can be narrowed to requiring that for the $x^2$, the x must be even (so that the number it follows is at least odd). Other examples?

Thanks!

2012ssohn
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math1234567
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    Such primes are Mersenne primes, $x$ must be $2$, since $x^n-1 = (x-1)(x^{n-1} + x^{n-2} + \dotsc + x + 1)$, and $n$ must be a prime. – Daniel Fischer Feb 11 '14 at 22:56
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    Probably the OP refers to this question, which has been near the top of the SE "Hot Network Questions" list for the past couple days, so has gotten a lot of rubbernecking traffic (11965 views). Better that than batman curves I suppose, since it can lead to some nontrivial mathematics. – Bill Dubuque Feb 11 '14 at 23:04

2 Answers2

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Hint: $x^n-1$ is always divisible through $x-1$.

Lucian
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For any even power greater than 2, there are no prime numbers directly proceeding those powers (see For $n>2$ is no primes followed by even powers?).

Community
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kleineg
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