A famous theorem of Dirichlet says that infinitely many primes are of the form:$\alpha n+\beta$, but are there infinitely many of the form: $\alpha ^n+\beta$, where $\beta$ is even and $\alpha$ is prime to $\beta$? or of the form $\alpha!+\gamma$, where $\gamma$ is odd?
Out of mere curiosity has this question come, thus any help is greatly appreciated.
Infinitely many primes are of the form $an+b$, but how about $a^n+b$?
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0Thanks for your information. – 2012-06-17
3 Answers
There are relatively prime non-trivial $\alpha$ and $\beta$, with $\beta$ even, such that $\alpha^n +\beta$ is not prime if $n \ge 1$. Easy, let $\beta$ have decimal expansion that ends in $4$, and let $\alpha>1$ have decimal expansion that ends in $1$.
A more subtle class of examples is illustrated by $625^n+4$. For this one we use the algebraic identity $x^4+4=(x^2-2x+2)(x^2+2x+2)$ to prove compositeness.
For the factorial question, a necessary condition for primality if $\alpha \gt 1$ is $\gamma=\pm 1$. Unfortunately it is not known whether there are infinitely many primes of the form $n!\pm 1$.
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0The question of whether there are $\alpha$, $\beta$ such that $\alpha^n+\beta$ is always prime. – 2012-06-16
Numbers $n$ such that $n! - 1$ is prime is http://oeis.org/A002982. The list begins, 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040. Presumably the list is infinite, but it appears that no one has proved it.
Numbers $n$ such that $n! + 1$ is prime is http://oeis.org/A002981. The list begins, 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209. As before, presumably the list is infinite, but it appears that no one has proved it.
Many references are given at those two webpages.
if you are interested if there is infinite prime number of the form $n!+1$ for infinite many n,then first use some example $n=2$ then $n!+1=3$ is prime,for n=3,$n!+1=7$ but comes question who can calculate $n!$ for n=50 for example,so it is difficult to say if there is infinity number of prime of this form
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1I don't know who can calculate $n!$ for $n=50$, but I do know that someone managed to prove that $150209!+1$ is prime. There are some very clever people out there, and there is more than one way to prove that a number is prime. See http://primes.utm.edu/primes/page.php?id=102627 – 2012-06-18