PrimorialsDate: 10/15/2003 at 14:45:49 From: Ryan Subject: the proof of infinite primes So we know that p_1 * p_2 * ... * p_n + 1 is either prime or divisible by a prime not included in the list. I can't find an example where the result is NOT prime, which makes me wonder whether the second part ("is divisible by a prime not included in the list") is superfluous. Do you know an example of one? Date: 10/15/2003 at 16:58:11 From: Doctor Peterson Subject: Re: the proof of infinite primes Hi, Ryan. We should first note that it isn't necessary that this ever actually happens; it's just part of the logic of the proof that we can't show that p_1 * p_2 * ... * p_n + 1 must be prime, but we _can_ show that if it isn't itself a new prime, it must be divisible by a new prime. The proof is perfectly valid even if we can never find any case where the latter is true. But it's not hard to find one, if you have the patience or technology to factor big numbers. Here are my results: 2+1 = 3 prime 2*3+1 = 7 prime 2*3*5+1 = 31 prime 2*3*5*7+1 211 prime 2*3*5*7*11+1 = 2311 prime 2*3*5*7*11*13+1 = 30031 = 59*509 You can see a list of which numbers of this form ARE prime here: Primorial http://mathworld.wolfram.com/Primorial.html p# + 1 is known to be prime for the primes p = 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, ... where p# means the product of all primes up to and including p. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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