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I made a conjecture about the values of n for which $3^n - 2^n$ is not prime, but I didn't succeed in proving the conjecture. My conjecture is the following: "Suppose n is an integer greater than or equal to 6. Then $3^n - 2^n$ is not prime". I tried to prove that by induction, but I got stuck. Any ideas?

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    Cameron Buie: Important remark! Byron Schmuland: Nice webpage! Thank you guys.2012-06-30

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Unfortunately, $3^{17}-2^{17}$ is prime.

It is clear that for primality the exponent $n$ must be prime. I would expect that no one knows whether there are infinitely many such primes. (This is based on the so far unsuccessful attempts to prove that there are infinitely many primes of the shape $2^n-1$.)

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    @Stavros: I do not know. Probably nobody does, if we consider the history of similar problems, like the Mersenne prime problem.2012-06-30
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Nope, $3^{31} - 2^{31} = 617671248800299$ which as everyone knows is prime.

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    Heeeeeeeeeee!!!!2012-07-01
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$n=17$ gives us $3^n-2^n = 129009091$, which is a prime number. Below is a list for $n \leq 100000$. $ \begin{array}{|c|c|} \hline n& (3^n-2^n)\\ \hline 2 & 5\\ 3 & 19\\ 5 & 211\\ 17 & 129009091\\ 29 & 68629840493971\\ 31 & 617671248800299\\ 53 & 19383245658672820642055731\\ 59 & 14130386091162273752461387579\\ 101 & 1546132562196033990574082188840405015112916155251\\ 277 & \text{A $133$ digit prime number}\\ 647 & \text{A $309$ digit prime number}\\ 1061 & \text{A $507$ digit prime number}\\ 2381 & \text{A $1137$ digit prime number}\\ 2833 & \text{A $1352$ digit prime number}\\ 3613 & \text{A $1724$ digit prime number}\\ 3853 & \text{A $1839$ digit prime number}\\ 3929 & \text{A $1875$ digit prime number}\\ 5297 & \text{A $2528$ digit prime number}\\ 7417 & \text{A $3539$ digit prime number}\\ 90217 & \text{A $43045$ digit prime number}\\ \hline \end{array}$ It is clear that if $3^n - 2^n$ is a prime, then $n$ has to be a prime but the converse is not true. In general, questions like these are likely to be hard.

For instance, replacing the $2$ by $1$ and $3$ by $2$ in your question, we get the Mersenne prime i.e. a prime of the form $2^n-1$. For instance, $31$ is a Mersenne prime, since $31 = 2^5-1$. Similarly, $127 = 2^7-1$ is also a Mersenne prime.

The Mersenne prime conjecture asks the following question

$\text{"Are there infinitely many Mersenne primes?"}$

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    @Stavros These questions tend to be extremely hard. I believe it's also unknown that $2^p - 1$ is _composite_ for infinitely many primes $p$, even though this is by far the more common occurrence (empirically). Presumably this is also true for $3^p-2^p$.2012-06-30