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

a1a2a3 . . . anan . . . a3a2a1    or a1a2a3 . . . an-1anan-1 . . . a3a2a1

where all a1, a2, a3, . . . an 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.

anan-1 . . . a2a1a0

and all numbers of the form

(k)an-1 . . . a1(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.

anan-1 . . . a2a1a0

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 an = a1 and an-1 = a0, then the number is a GSP. Since the GSPs where an = a0 have already been counted in the previous step, the conditions are

an = a1 and an-1 = a0     and an ≠ a0

The situation is equivalent to choosing a nonzero digit for an, and decimal digits for an-1 and a0 that satisfy these conditions. This probability of this is easy to compute and is 0.009.

If an = a2, an-1 = a1 and an-2 = a0, then the number is a GSP. To determine the probability here, we need to choose six digits, where an 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 an = a3, an-1 = a2, an-2 = a1 and an-3 = a0 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.

# References

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

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