Factors of primorials

[grandpascorpion]

Just curious,

The lowest factor of 2* 3 * 5 * 7 * 11 * 13 * 17 +1

is 19, the next prime after 17.

Does anyone know of other cases like this?

[Mystwalker]

Sounds like the Law of Small Numbers.

[grandpascorpion]

Maybe a law of one?

I have checked thru 227. This is the only guy I have seen so far.

[Yogi]

What about 2 + 1 is divisable by 3?

Originally Posted by grandpascorpion

Does anyone know of other cases like this?

The only examples I know are 1459 and 2999 (checked through 2454587).

[grandpascorpion]

I was thinking of factors only and missed the obvious I guess.

Thanks for the other values. Could you point me to the link where you saw them?

-Grandpa

[Yogi]

Not really, since I used a self written program. I attached it (in case you want to see it). I know it's not optimal, but it works.

And before I forget: You'll need NTL (Number theoretic library from Victor Shoup; free) if you want to compile it. I couldn't attach the zipped (cygwin) .exe, since it's still too large.

Usage:
prim <n>

where n denotes the n-th prime, so you'd get the result

(p(1)# + 1) % p(2) = 0

if you run it with "prim 0". (It always skips the first n-value; there is no p(0))

[grandpascorpion]

Thanks Yogi

[Ken_g6]

This sounds alot like Euclid's proof that there are infinitely prime numbers. I believe it says that n primorial + 1 is not divisible by any prime smaller than n, correct?

[jinydu]

Originally Posted by Ken_g6

This sounds alot like Euclid's proof that there are infinitely prime numbers. I believe it says that n primorial + 1 is not divisible by any prime smaller than n, correct?

Yes, but not all numbers of the form (n primorial) + 1 are prime because you can't rule out factors larger than n.

[maxal]

Originally Posted by Yogi

What about 2 + 1 is divisable by 3?

The only examples I know are 1459 and 2999 (checked through 2454587).

These are the only known such primes. See http://www.research.att.com/projects/OEIS?Anum=A058233