Conjecture: If p1 * p2 * p3 * ... pn + 1 is prime for every positive integer n, where p1,p2,...,pn are the n smallest prime number
So far, I have tried this
Proof (direct) :
Let pj be a prime number which is greater than 1
(p1*p2*p3*p4*...pn) % pj = 0
1 is divisible by pj
Therefore this conjecture is true?
Not by any prime in your list used in constructing the number, yes. If that's what you are trying to prove then the proof is correct. As in CaptainBlack's example
Note that 30031 is not divisible by any of 2, 3, 5, 7, 11, or 13, but it is divisible by the primes 59 and 509.
-Dan
This is how I understand this user.
Let be the first primes.
Then, is not divisible by any of these . (Remember Euclid's Proof).
So, (and that is what he claims) it must mean that is prime itself because it is not divisible by any of those primes.
That is his conjecture.
But that is false because we only showed that is divisible by another prime. Either itself, if is it, or a completely different prime if it is not.