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Old 2005-01-20, 21:27   #1
grandpascorpion
 
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Default Factors of primorials

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?
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Old 2005-01-20, 21:53   #2
Mystwalker
 
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Sounds like the Law of Small Numbers.
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Old 2005-01-20, 22:17   #3
grandpascorpion
 
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Maybe a law of one?

I have checked thru 227. This is the only guy I have seen so far.
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Old 2005-01-22, 08:26   #4
Yogi
 
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What about 2 + 1 is divisable by 3?

Quote:
Originally Posted by grandpascorpion
Does anyone know of other cases like this?
The only examples I know are 1459 and 2999 (checked through 2454587).
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Old 2005-01-23, 14:00   #5
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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
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Old 2005-01-23, 17:14   #6
Yogi
 
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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))
Attached Files
File Type: zip prim.zip (599 Bytes, 65 views)
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Old 2005-01-25, 00:26   #7
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Thanks Yogi
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Old 2005-02-06, 06:06   #8
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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?
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Old 2005-02-06, 06:41   #9
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Quote:
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.
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Old 2005-02-10, 07:13   #10
maxal
 
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Quote:
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
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