# 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

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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