Problems & Puzzles: Puzzles Puzzle 131.
Growing primes For sure you know the primes 73939133 and 357686312646216567629137. They are respectively the largest righttruncatable and the largest lefttruncatable primes. These primes are the matter of two articles (1, 2) of the well known glossary prime pages of Chris Caldwell, and was also part of the matter of an old puzzle in my pages. Let's concentrate by the moment in the righttruncatable record prime. 73939133 is a prime & the largest ever possible prime such that all of the substrings of the original prime are also prime numbers 73939133 In the original definition you choose an initial prime that remains prime after cutting one digit at a time "by the right" until you reach to a onedigit prime. But you can view this process upside down: 73939133 is the largest prime that can be constructed over an initial onedigit prime adding one digit by the right producing in every step of the process only prime numbers 7 From this point of view you can see this sequences as a sequence of "growing primes" (by the right). This is the point of view of a puzzle posted by Frank Rubin in his always interesting contest pages (The Contest Center). But this point of view permits to start not from a onedigit primes (that is to say 2, 3, 5 or 7) but you can start with any prime number. For example: the prime 43 is the smallest prime that can grow 5 times: 43 From this point of view 73939133 is no more a limit to the growth and you may have a sequence of more than 8 growing primes As a matter of fact Rubin makes the following puzzling statements about this kind of objects: 1)
"no matter what prime you start with and what digits you append
each such sequence always ends. That is, there comes a point where there
is no digit that can be appended to form another prime." 2)
"Conversely ... that for any N there is always a sequence of N
such primes [starting with a proper prime]" Questions: 1) Can you prove the two Rubin's statements about the growing primes by the right? 2) Can you extend the following table
3) Remain the validity of the Rubin's statements for "growing primes" by the left? 4) Construct the corresponding Table for the growing primes by the left.
Solution Jud McCranie comments (24/3/01): "It is easy to see that at most one of the digit of 1 and 7 can be added. Except for possibly one 1 or one 7, the rest of the digits added to the right must be 3 or 9" *** Later ( 25/3/01) he added: I get this for part 4:
*** On April 13, 2007 J. K. Andersen wrote: 2) K Least initial Largest growing (by the prime right) prime    12 2744903797 2744903797:739993993333 13 5232781111 5232781111:9399399313393 14 133028062963 133028062963:79339933399333 4) K Least Largest growing (by the left) prime prime    0 2 :2 1 1303 9:1303 2 757 67:757 3 379 576:379 4 1049 6798:1049 5 139 84954:139 6 1061 939135:1061 7 383 7213272:383 8 359 96212633:359 9 109 997843951:109 10 223 9993243381:223 11 107 69213573395:107 12 227 976263648789:227 13 23 8372669942975:23 14 43 69334276539184:43 15 31 153994591539733:31 16 13 1986153454518136:13 17 11 48616876513572342:11 18 29 991273987576686678:29 19 3 3648495721353667688:3 20 19 98633312193846876837:19 21 59 787383153426187623456:59 22 37 3576863126462165676291:37 23 7 35768631264621656762913:7 24 76922083307 968429465126331891833378:76922083307 If the least prime can grow to more than one prime of the same length, then the largest of those primes is listed. Jud McCranie listed a smaller prime of the same length for K = 4, 5, 7, 9, 10, 12. The solutions for K=22 and K=23 grow to the largest lefttruncatable prime. It was first computed at least 30 years ago: http://links.jstor.org/sici?sici=00255718%28197701%2931%3A137%3C265%3AOTP%3E2 "On Truncatable Primes" by I. O. Angell and H. J. Godwin, Mathematics of Computation, Vol. 31, No. 137 (Jan., 1977), pp. 265267 The solution for K=24 is the only 11digit prime which can grow 24 times, so 96842946512633189183337876922083307 is the smallest (and only known) prime which can be truncated more times than the largest lefttruncatable prime. "...what can you say about the Rubin's statement?:"
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