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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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