Problems & Puzzles: Puzzles

 Puzzle 339. Primes & persistence. For sure you already know what 'persistence' means. Now I will switch a little bit the original concept. I will call 'prime persistence +1' to the length (k) of a sequence of prime numbers p1, p2, ...pk, such that a) p1 is a starting prime number and b) every prime pi, i>1, is the product of the digits of the previous prime plus one c) pk is a one digit prime. Example: k=5+ : 1569889  155521  251  11  2 An example of 'prime persistence -1' of length k is the following one: k=7- : 158888689  8847359  241919  647  167  41  3 Question: Find larger sequences of both type.

Wilfred Whiteside sent (December, 2005) the following results:

I tried out puzzle 339 (primes and persistence).  I tried sieving to 72x10^9 as a lazy try.  Nothing.  Then I installed a big number package on my new PC and learned how to use it (I hope).  Then I tried the k+ case using the fact that p2 must be a product of single digits +1, so p2-1 must be a product of powers of 2,3,5,7.  I tried all products up to 140 digits with powers of 2 up to 280, powers of 3 up to 200, powers of 5 up to 110, and powers of 7 up to 110.  I could find nothing to beat the k5+ example.  I could easily have a bug in my code, but it did find the k5+ example almost instantly, so that was a good sign.

Then I ran the k- case, expecting to find nothing good.  But it easily spit out
k8-: 35557777777987, 155649626999, 283435199, 233279, 2267, 167, 41, 3

... I ran a final run on puz339 searching for a k9- case.  I considered all values of p2 for numbers up to 140 digits that were a product of powers of 2 up to 280 and powers of 3 up to 200 and powers of 5 up to 110 and powers of 7 up to 150.  Nothing worked.  So if my code is valid, then a k9- sequence would have a very large value if it exists (at least greater than 80 digits and probably much larger).

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