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I noticed that $41$ plus primorials is prime when I add $2$ to $41=43$, then $6$ to $41= 47$, then $30+41=71$ all the way up to $41+9699690=9699731$. Hence I can add the first $8$ primorials to get primes. Do you know if there is a different prime that will allow the addition of more than the first $8$ primorials?

Leyla Alkan
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J. M. Bergot
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  • My first instinct is to encourage you to simply write a small program to enumerate examples. For reasonably "small" numbers (which can actually be quite big) you should be able to simple run primality tests until they work, or you get bored. – Valborg Mar 23 '18 at 20:03
  • I checked at prime curios and nothing there about number 41 doing this. If no one knows, then so be it. This will allow a more thorough search than I can do. – J. M. Bergot Mar 23 '18 at 20:07
  • I don't know what prime curios is, but this probably doesn't have to do with the number 41. There are likely many small examples we can find; I am simply encouraging you to go find them. – Valborg Mar 23 '18 at 20:10
  • I'm surprised! I thought everyone knew of this website. Here it is :https://primes.utm.edu/curios/page.php/41.html It is still a lovely problem for those who enjoy toying with a LARGE computer. – J. M. Bergot Mar 23 '18 at 20:13
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    Possibly related $n^2+n+41$ is prime for $n=0,1,\cdots, 39$. This dates back to Euler 1772. – Malcolm Mar 23 '18 at 20:21

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I just wrote a little Maple script to find some examples of primorially separated primes, as I will call them. Here are some results:

p=3 is the smallest prime which has a run of 1 primorially separated primes after it.

p=5 is the smallest prime which has a run of 2 primorially separated primes after it.

p=11 is the smallest prime which has a run of 3 primorially separated primes after it.

p=17 is the smallest prime which has a run of 4 primorially separated primes after it.

p=41 is the smallest prime which has a run of 8 primorially separated primes after it.

p=86351 is the smallest prime which has a run of 10 primorially separated primes after it.

Finding a run of 11 or more is turning out to take quite some time, but if my script finds anything I will update further.

Valborg
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  • Certainly daring boldness to actually look for some numbers rather than merely talking Grand Theory. ThanXXXXX gobs! Maybe this could be the start of some recreational experimental computational derring-do? – J. M. Bergot Mar 24 '18 at 17:28
  • p=235313357 is the smallest prime which has a run of 11. – pietfermat May 17 '18 at 11:33