The Density of Generalized Smarandache Palindromes

Charles Ashbacher

Charles Ashbacher
Technologies

Hiawatha, IA 52233 USA

Lori Neirynck

Mount Mercy College

1330 Elmhurst Drive

Cedar Rapids, IA 52402 USA

An integer is said to be a
palindrome if it reads the same forwards and backwards. For example, 12321 is a
palindromic number. It is easy to prove that the density of the palindromes is
zero in the set of positive integers.

A Generalized Smarandache Palindrome (GSP) is
any integer of the form

a_{1}a_{2}a_{3} . .
. a_{n}a_{n} . . . a_{3}a_{2}a_{1 } or a_{1}a_{2}a_{3}
. . . a_{n-1}a_{n}a_{n-1} . . . a_{3}a_{2}a_{1}

where
all a_{1, }a_{2, }a_{3,} . . . a_{n} are
integers having one or more digits [1], [2]. For example,

10101010 and
101010

are
GSPs because they can be split into the forms

(10)(10)(10)(10)
and (10)(10)(10)

and
the segments are pairwise identical across the middle of the number.

As a
point of clarification, we remove the possibility of the trivial case of
enclosing the entire number

12345
written as (12345)

which
would make every number a GSP. This possibility is eliminated by requiring that
each number be split into at least two segments if it is not a regular
palindrome.

Also,
the number 100610

is
considered to be a GSP, as the splitting

(10)(06)(10)

leads
to an interior string that is a separate segment, which is a palindrome by
default.

Obviously, since each regular palindrome is
also a GSP and there are GSPs that are not regular palindromes, there are more
GSPs than there are regular palindromes. Therefore, the density of GSPs is
greater than or equal to zero and we consider the following question.

What
is the density of GSPs in the positive integers?

The
first step in the process is very easy to prove.

**Theorem:** The density of GSPs in the
positive integers is greater than 0.1.

Proof:
Consider a positive integer having an arbitrary number of digits.

a_{n}a_{n-1} . . .
a_{2}a_{1}a_{0}

and
all numbers of the form

(k)a_{n-1} . . . a_{1}(k)

are
GSPs, and there are nine different choices for k. For each of these choices,
one tenth of the values of the trailing digit would match it. Therefore, the density
of GSPs is at least one tenth.

The
simple proof of the previous theorem illustrates the basic idea that if the
initial and terminal segments of the number are equal, then the number is a
generalized palindrome and the values of the interior digits are irrelevant.
This leads us to our general theorem.

**Theorem: **The density of GSPs in the
positive integers is approximately 0.11.

Proof:
Consider a positive integer having an arbitrary number of digits.

a_{n}a_{n-1}
. . . a_{2}a_{1}a_{0}

If
the first and last digits are equal and nonzero, then the number is a
generalized palindrome. As was demonstrated in the previous theorem, the
likelihood of this is 0.10.

If
a_{n} = a_{1} and a_{n-1} = a_{0}, then the
number is a GSP. Since the GSPs where a_{n} = a_{0} have already
been counted in the previous step, the conditions are

a_{n} = a_{1} and a_{n-1}
= a_{0 }and a_{n}
≠ a_{0}

The
situation is equivalent to choosing a nonzero digit for a_{n}, and
decimal digits for a_{n-1} and a_{0} that satisfy these conditions.
This probability of this is easy to compute and is 0.009.

If
a_{n} = a_{2}, a_{n-1} = a_{1} and a_{n-2}
= a_{0}, then the number is a GSP. To determine the probability here,
we need to choose six digits, where a_{n} is nonzero and the digits do
not also satisfy the conditions of the two previous cases. This is also easily
computed, and the value is 0.0009.

The
case where a_{n} = a_{3}, a_{n-1} = a_{2}, a_{n-2}
= a_{1} and a_{n-3} = a_{0} is the next one, and the
probability of satisfying this case after failing in the three previous cases
is 0.0000891.

The
sum of these probabilities is 0.10 +
0.009 + 0.0009 + 0.0000891, which is 0.1099891.

This
process could be continued for initial and terminal segments longer and longer,
but the probabilities would not be enough to make the sum 0.11.

1.
G.
Gregory, Generalized Smarandache Palindromes, http://fs.unm.edu/GSP.htm.

2.
F.
Smarandache, Generalized Palindromes, Arizona State University Special
Collections, Tempe.