generalized Smarandache palindrome

generalized Smarandache palindrome

A

generalized Smarandache palindrome

(GSP) is a concatenated number of the form:

a

1

⁢

a

2

⁢

…

⁢

a

n

⁢

a

n

⁢

…

⁢

a

2

⁢

a

1

, for

n

≥

1

, or

a

1

⁢

a

2

⁢

…

⁢

a

n

-

1

⁢

a

n

⁢

a

n

-

1

⁢

…

⁢

a

2

⁢

a

1

, for

n

≥

2

, where all

a

1

,

a

2

,

…

,

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

∈

{

1

,

2

,

…

,

9

}

and

b

∈

{

0

,

1

,

2

,

…

,

9

}

. There are

9

×

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

∈

{

1

,

2

,

…

,

9

}

and

B

∈

{

00

,

01

,

02

,

03

,

…

,

99

}

-

{

00

,

11

,

22

,

33

,

…

,

99

}

;
2) or CC, where

C

∈

{

10

,

11

,

12

,

…

,

99

}

-

{

11

,

22

,

33

,

…

,

99

}

.
In the first case, one has

9

×

(

100

-

10

)

=

9

×

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.htm

The Density of Generalized Smarandache Palindromes, Journal of Recreational Mathematics, Vol. 33 (2), 2006

2. G. Gregory,

http://www.gallup.unm.edu/ smarandache/GSP.htm

Generalized Smarandache Palindromes

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

4. M. Khoshnevisan, Proposed Problem

♯

1062 (on Generalized Smarandache Palindrome), The

Π

ME Epsilon, USA, Vol. 11, No. 9, p. 501, Fall 2003.

5. Mark Evans, Mike Pinter, Carl Libis, Solutions to Problem

♯

1062 (on Generalized Smarandache Palindrome), The

Π

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=A082461

Encyclopedia of

Integers

, Sequence A082461

7. F. Smarandache,

http://www.gallup.unm.edu/ smarandache/Sequences-book.pdf

Sequences of Numbers Involved in Unsolved Problems, Hexis, 1990, 2006

Title

generalized Smarandache palindrome

Canonical

name

GeneralizedSmarandachePalindrome

Date of creation

2013-03-22 17:03:25

Last modified on

2013-03-22 17:03:25

Owner

dankomed (17058)

Last modified by

dankomed (17058)

Numerical id

7

Author

dankomed (17058)

Entry type

Definition

Classification

msc 11Z05

Related topic

FlorentinSmarandache

Generated on Fri Feb 9 15:47:51 2018 by

LaTeXML