[seqfan] Shortest sequence to get prime

[Alonso Del Arte]

As we all know, there are infinitely many primes, and most of us here know
at least Euclid's proof and one other proof.

Given a positive prime *p*, what is is the shortest, non-empty finite list
of the smallest distinct positive primes that does not contain *p* such
that the product of those distinct primes is one less than a multiple of *p*?
As you can see from the following list, for some primes I run into a little
bit of a conundrum.

  3 {2}
  5 {2, 7}
  7 {2, 3}
 11 {2, 5}
 13 {2, 19} OR {3, 17}?
 17 {3, 11}
 19 {37}
 23 {2, 11}
 29 {3, 19}
 31 {2, 3, 5} OR {61}?
 37 {73}
 41 {2, 61}
 43 {2, 3, 7} OR {5, 17}?
 47 {2, 23}
 53 {3, 5, 7} OR {2, 79}?
 59 {2, 29}
 61 {2, 7, 13}
 67 {2, 3, 11} OR {7, 19}?
 71 {2, 5, 7} OR {3, 47}?
 73 {5, 29}
 79 {2, 3, 13} OR {157}?
 83 {2, 41}
 89 {3, 59}
 97 {193}

Maybe I should choose the list with the smaller sum, so for 31, {2, 3, 5}
is preferred to {61}. Though I feel like maybe {2, 19} should nonetheless
be preferred to {3, 17} for 13.

Al

-- 
Alonso del Arte
Author at SmashWords.com
<https://www.smashwords.com/profile/view/AlonsoDelarte>
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