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Pairs of primes (Posted on 2019-07-27) Difficulty: 4 of 5
Imagine a bag containing cards representing all n-digit odd numbers. A random card is drawn and two new numbers are created by preceding the drawn number by each of its even neighbors.

What is the probability that each of those 2 numbers is prime?

For n=1 there are 5 cards i.e. 1,3,5,7 and 9. Clearly only numbers 3 and 9 qualifiy since fboth 23 and 43 are primes and so are 89 and 109 & there are no other answers. So for n=1 p=0.4 is the probability we were looking for.
For n=2 I will not provide the answer but will show you one of the qualifying numbers e.g. 69, since both 6869 and 7069 are prime.

Now evaluate the correct probabilities for n=2,3, ...8,9 (or as far as your resources allow) - and you will get a sequence for which you may be credited @ OEIS.

So this time you get a task both challenging and rewarding!

No Solution Yet Submitted by Ady TZIDON    
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Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): via computer - the start of the soln | Comment 9 of 10 |
(In reply to re: via computer - the start of the soln by Charlie)

Thanks to Charlie for spotting the bug in my program. I have fixed it and my numbers agree with his. I also used the tricks suggested by xdog which speeds things up a bit. But, without downloading a huge list of primes, I also only get to i=6 in a minute of so, and then the (2i)^2 nature of the problem grinds the show down.

Edited on July 30, 2019, 7:51 pm
  Posted by Steven Lord on 2019-07-30 17:00:40

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