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Does it continue? 4: Euclid's next prime (Posted on 2017-09-21) Difficulty: 3 of 5
Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Let P(n)= The product of the first n primes.

P(n)+1 is like Euclid's method to show there are infinitely many primes, and may or not be prime itself. Now look at the difference between P(n) and the next prime after P(n)+1.

n=1, 5-(2)=3
n=2, 11-(2x3)=5
n=3, 37-(2x3x5)=7
n=4, 223-(2x3x5x7)=13
n=5, 2333-(2x3x5x7x11)=23
n=6, 30047-(2x3x5x7x11x13)=17
n=7, 510529-(2x3x5x7x11x13x17)=19
n=8, 9699713-(2x3x5x7x11x13x17x19)=23

Are these differences always prime?

No Solution Yet Submitted by Jer    
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Your thoughts? | Comment 2 of 3 |
The sequence is easily looked up, but don't without sharing your opinion.  

A hint: in 1988, the date of Richard K. Guy's paper, the question was unanswered.

  Posted by Jer on 2017-09-21 21:05:31
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