Problems & Puzzles: Puzzles

Puzzle 70. Primes Double Tree (a puzzle suggested by Paul Leyland)

"The 1-digit primes are 2, 3, 5, 7 and from these we can generate two classes of 2-digit primes: (23, 29, 31, 37, 53, 59, 71, 73, 79) and (13, 23, 43, 53, 73, 83,  17, 37, 47, 67, 97) according to whether the other digit is placed before or after the first.  These two classes can be extended in the obvious way to longer lengths.  Eventually, it is no longer possible to create primes in this manner…" (Paul Leyland, 27/09/99).

In base 10, I have obtained the number of primes in each generation until certain extent.

Quantity of primes per Generation, Base 10

 Digits added Generation (number of digits) Before After 1 (initial) 4 4 2 11 9 3 39 14 4 99 16 5 192 15 6 326 12 7 ??? 8 8 ??? 5 9 ??? 0 (end) 10 ??? Total = 83

Questions:

1. Can you complete the table for Base 10?
2. Can you obtain the corresponding tables to all the bases less than 10?

Solutions

Eric Weisstein solved 1. of this puzzle. Here is his e-mail (4/10/99):

"...here is the complete count of left prime strings less than 10^n [and greater than 10^(n-1)]: 4, 11, 39, 99, 192, 326, 429, 521, 545, 517, 448, 354, 276, 212, 117, 72, 42, 24, 13, 6, 5, 4, 3, 1, 0. total: 4260
Reference:  www.treasure-troves.com/math/PrimeString.html

According to the sent reference this problem was partially (the base 10) solved previously. In particular Baillie in1995 obtained the largest prime of the sequence "digits added before". J. Shallit (when?) got another thoerical interesting result. But please see all the details in the Eric's site.

***

Martin Renner wrote (March 25, 2005):

Solution 2. Tables to all the bases less than 10.

 Lefttruncatable primes base digits 3 4 5 6 7 8 9 1 1 2 2 3 3 4 4 2 1 2 4 4 6 12 9 3 1 3 4 12 6 29 15 4 3 3 25 4 50 17 5 3 1 44 1 66 24 6 3 1 54 1 77 16 7 60 1 61 9 8 62 51 6 9 59 38 5 10 51 27 3 11 35 17 12 20 8 13 12 3 14 7 2 15 3 1 16 2 17 1 sum 3 16 15 454 22 446 108

 Righttruncatable primes base digits 3 4 5 6 7 8 9 1 1 2 2 3 3 4 4 2 1 2 4 5 5 10 8 3 1 2 4 6 6 14 13 4 1 1 3 8 4 16 14 5 1 9 1 13 13 6 4 7 4 7 1 3 4 8 1 3 9 2 10 3 sum 4 7 14 36 19 68 68

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