The Smarandache P and S persistence of a prime

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in this article there are introduced two new concepts: the Smarandache P-persistence and the Smarandache S-persistence of a prime number.
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  Scientia Magna Vol. 1 (2006), No. 2, 71-75 The Smarandache  P   and  S   persistence of aprime Felice Russo Micron Technology Italy67051 Avezzano (AQ), Italy In [1], Sloane has defined the multiplicative persistence of a number in the following man-ner. Let’s  N   be any  n -digits number with  N   =  x 1 x 2 x 3 ··· x n  in base 10. Multiplying togetherthe digits of that number ( x 1 · x 2 ····· x n ), another number  N   results. If this process is iterated,eventually a single digit number will be produced. The number of steps to reach a single digitnumber is referred to as the persistence of the srcinal number  N  . Here is an example:679 → 378 → 168 → 48 → 32 → 6 . In this case, the persistence of 679 is 5.Of course, that concept can be extended to any base  b . In [1], Sloane conjectured that, inbase 10, there is a number  c  such that no number has persistence greater than  c . According to acomputer search no number smaller than 10 50 with persistence greater than 11 has been found.In [2], Hinden defined in a similar way the additive persistence of a number where, instead of multiplication, the addition of the digits of a number is considered. For example, the additivepersistence of 679 is equal to 2.679 → 22 → 4 . Following the same spirit, in this article we introduce two new concepts: the Smarandache P  -persistence and the Smarandache  S  -persistence of a prime number. Let  X   be any  n -digitsprime number and suppose that  X   =  x 1 x 2 x 3 ··· x n  in base 10. If we multiply together thedigits of that prime ( x 1 · x 2 ····· x n ) and add them to the srcinal prime ( X   + x 1 · x 2 ····· x n )a new number results, which may be a prime. If it is a prime then the process will be iteratedotherwise not. The number of steps required to  X   to collapse in a composite number is calledthe Smarandache  P  -persistence of prime  X  . As an example, let’s calculate the Smarandache P  -persistence of the primes 43 and 23:43 → 55;23 → 29 → 47 → 75 , which is 1 and 3, respectively. Of course, the Smarandache  P  -persistence minus 1 is equalto the number of primes that we can generate starting with the srcinal prime  X  . Beforeproceeding, we must highlight that there will be a class of primes with an infinite Smarandache P  -persistence; that is, primes that will never collapse in a composite number. Let’s give an  72  Felice Russo No. 2 example:61 → 67 → 109 → 109 → 109 ···  . In this case, being the product of the digits of the prime 109 always zero, the prime 61 willnever reach a composite number. In this article, we shall not consider that class of primes sinceit is not interesting. The following table gives the smallest multidigit primes with Smarandache P  -persistence less than or equal to 8:Smarandache  P  -persistence Prime1 112 293 234 3475 2936 2397 574878 486193By looking in a greater detail at the above table, we can see that, for example, the secondterm of the sequence (29) is implicitly inside the chain generated by the prime 23. In fact:29 → 47 → 7523 → 29 → 47 → 75We can slightly modify the above table in order to avoid any prime that implicitly is insideother terms of the sequence.Smarandache  P  -persistence Prime1 112 1633 234 5635 14516 2397 574878 486193Now, for example, the prime 163 will generate a chain that isn’t already inside any otherchain generated by the primes listed in the above table. What about primes with Smarandache P  -persistence greater than 8? Is the above sequence infinite? We will try to give an answer  Vol. 1 The Smarandache  P   and  S   persistence of a prime  73to the above question by using a statistical approach. Let’s indicate with  L  the Smarandache P  -persistence of a prime. Thanks to an  u -basic code the occurrrencies of   L  for different valuesof   N   have been calculated. Here an example for  N   = 10 7 and  N   = 10 8 :Figure 1. Plot of the occurrencies for each  P  -persistence at two different values of   N  .The interpolating function for that family of curves is given by: a ( N  ) · e − b ( N  ) · L where  a ( n ) and  b ( n ) are two function of   N  . To determine the behaviour of those two functions,the values obtained interpolating the histogram of occurencies for different N have been used:N a b1 . 00 E   + 04 2238 . 8 1 . 31311 . 00 E   + 05 17408 1 . 43291 . 00 E   + 06 121216 1 . 53391 . 00 E   + 07 1 . 00 E   + 06 1 . 69911 . 00 E   + 08 1 . 00 E   + 07 1 . 968  74  Felice Russo No. 2 Figure 2. Plot of the two functions  a ( N  ) and  b ( N  ) versus  N  According to those data we can see that : a ( N  ) ≈ k · N b ( N  ) ≈ h · ln( N  ) + c where  k ,  h  and  c  are constants (see Figure 2).So the probability that  L ≥ M   (where  M   is any integer) for a fixed  N   is given by: P  ( L ≥ M  ) ≈    ∞ M   kN   · e − ( h ln N  + c ) · L dL    ∞ 0  kN   · e − ( h ln N  + c ) · L dL =  e − ( h · ln N  + c ) · M  and the counting function of the primes with Smarandache  P  -persistence  L  =  M   below  N   isgiven by  N   · P  ( L  =  M  ). In Figure 3, the plot of counting function versus  N   for 4 different  L values is reported. As we can see, for  L <  15 and  L ≥ 15 there is a breaking in the behaviour of the occurrencies. For  L ≥ 15, the number of primes is very very small (less than 1) regardlessthe value of   N   and it becomes even smaller as  N   increases. The experimental data seem tosupport that  L  cannot take any value and that most likely the maximum value should be  L  = 14or close to it. So the following conjecture can be posed: Conjecture 1.  There is an integer  M   such that no prime has a Smarandache P  -persistencegreater than  M  . In other words the maximum value of Smarandache  P  -persistence is finiteFigure 3. Counting function for the  P  -persistence for difference values of   N 
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