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Is there a greatest prime number? My intuition is that prime number series is a never ending. Can somebody explain whether or not greatest prime number exist?

user42461
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  • http://math.stackexchange.com/questions/50006/different-ways-to-prove-there-are-infinitely-many-primes – joeb Mar 16 '17 at 21:29
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    There are infinitely many prime numbers (hence no largest). Look up Euclid's proof (search terms: "primes", "Euclid"). – quasi Mar 16 '17 at 21:30
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    If you don't want to read other people's answers and want to prove it on your own, think about $\gcd(a,an+1)$ and suppose that there was a finite list of primes... – JMoravitz Mar 16 '17 at 21:31
  • As an aside, The very first chapter of proofs from THE BOOK gives multiple proofs of why there are infinitely many prime numbers. – JMoravitz Mar 16 '17 at 21:41
  • I got it. Thanks. – user42461 Mar 16 '17 at 21:42
  • Dirichlet even proved that every arithmetic progression $an+b$ , $a,b$ positive integers $n=1,2,3,\cdots $ contains infinite many primes, if $a$ and $b$ are coprime. – Peter Mar 16 '17 at 22:09
  • If you only care that it be prime without putting any other requirements on it, yeah, there are infinitely many primes. Go to the library, look around QA 245 (or around 510.1 Dewey), pick any book about prime numbers, you'll probably find at least one proof, if not an entire first chapter of proofs. – Robert Soupe Mar 17 '17 at 01:57

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Greatest prime number does not exist. Simplest proof: Let us have finite number of primes. Then we could list them all : $\{a_1,...,a_n\}$. Then all other could be divided with no remains with one of them, or if you like more $(\forall a)(\exists k, 1 \leq k \leq n)a_k|a$. But, consider $a_1....a_n+1$, it will not be in the first set, and would not satisfy the condition, hence is a prime , contradiction!

nikola
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    Are you saying that if $a_{n}$ is the largest prime, $a_{n}+1$ is a prime and hence we arrive at a contradiction? Because that would seem flawed. – mrnovice Mar 16 '17 at 21:57
  • @mrnovice The notation could be clearer. What nikola meant is multiply $a_1$ by $a_2$ by all the others up to $a_n$ and then add 1. For example, let's say 2, 3, 5, 7 are the only primes. Multiply them together to get 210, add 1 to that, that's 211, which is either a new prime or is composite but divisible by a prime not on our list. – Robert Soupe Mar 17 '17 at 01:51
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    @RobertSoupe Ah okay, I agree with his answer then. Although it would have been a bit clearer to use: $\prod_{i=1}^{n}a_{i} + 1$ – mrnovice Mar 17 '17 at 01:53
  • @mrnovice That's probably what I would have used. – Robert Soupe Mar 17 '17 at 02:00