# generalized Smarandache palindrome

A generalized Smarandache palindrome (GSP) is a concatenated number of the form: $a_{1}a_{2}\ldots a_{n}a_{n}\ldots a_{2}a_{1}$, for $n\geq 1$, or $a_{1}a_{2}\ldots a_{n-1}a_{n}a_{n-1}\ldots a_{2}a_{1}$, for $n\geq 2$, where all $a_{1},a_{2},\ldots,a_{n}$ are positive integers of various number of digits in a given base $b$.

Proposed Problem

Find the number of GSP of four digits that are not palindromic numbers in base 10.

M. Khoshnevisan, Griffith University, Gold Coast, Queensland 9726, Australia.

Solution

Before solving the problem, let see some examples:

1) 1235656312 is a GSP because we can group it as (12)(3)(56)(56)(3)(12), i.e. ABCCBA.

2) The number 5675 is also a GSP because it can be written as (5)(67)(5).

3) Obviously, any palindromic number is a GSP number as well.

A palindromic number of four digits has the concatenated form: abba, where $a\in\{1,2,\ldots,9\}$ and $b\in\{0,1,2,\ldots,9\}$. There are $9\times 10=90$ palindromic numbers of four digits. For example, 1551, or 2002 are palindromic (and, of course, GSP too); yet 3753 is not palindromic but it is a GSP for 3753=3(75)3, i.e. of the form ABA; similarly 4646, for it can be organized as (46)(46), i.e. of the form CC. Therefore, a SGP, different from a palindromic number, should have the concatenated forms: 1) ABA, where $A\in\{1,2,\ldots,9\}$ and $B\in\{00,01,02,03,\ldots,99\}-\{00,11,22,33,\ldots,99\}$; 2) or CC, where $C\in\{10,11,12,\ldots,99\}-\{11,22,33,\ldots,99\}$. In the first case, one has $9\times(100-10)=9\times 90=810$. In the second case, one has $90-9=81$. Total: $810+81=891$ GSP numbers of four digits which are not palindromic.

References

1. Charles Ashbacher, Lori Neirynck, www.gallup.unm.edu/ smarandache/GeneralizedPalindromes.htmThe Density of Generalized Smarandache Palindromes, Journal of Recreational Mathematics, Vol. 33 (2), 2006

2. G. Gregory, http://www.gallup.unm.edu/ smarandache/GSP.htmGeneralized Smarandache Palindromes

3. M. Khoshnevisan, ”Generalized Smarandache Palindrome”, Mathematics Magazine, Aurora, Canada, 10/2003.

4. M. Khoshnevisan, Proposed Problem $\sharp$1062 (on Generalized Smarandache Palindrome), The $\Pi$ME Epsilon, USA, Vol. 11, No. 9, p. 501, Fall 2003.

5. Mark Evans, Mike Pinter, Carl Libis, Solutions to Problem $\sharp$1062 (on Generalized Smarandache Palindrome), The $\Pi$ME Epsilon, Vol. 12, No. 1, 54-55, Fall 2004.

6. N. Sloane, http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A082461Encyclopedia of Integers, Sequence A082461

7. F. Smarandache, http://www.gallup.unm.edu/ smarandache/Sequences-book.pdfSequences of Numbers Involved in Unsolved Problems, Hexis, 1990, 2006

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