Palindrome Studies (Part 1) The Palindrome Concept and Its Applications to Prime Numbers Hen ry Ibstedt

Abstract: This article ongmates from a proposal by M. L. Perez of American Research Press to carry out a study on Smarandache generalized palindromes [1]. The prime numbers were chosen as a fIrst)set of numbers to apply the development of ideas and computer programs on. The study begins by exploring regular pritlle number palindromes. To continue the study it proved useful to introduce a new concept, that of extended palindromes with the property that the union of regular palindromes and extended palindromes form the set of Smarandache generalized palindromes. An interesting observation is proved in the article, namely that the only regular prime number palindrome with an even number of digits is 11. 1. Regular Palindromes Definition: A positive integer is a palindrome if it reads the same way forwards and backwards. Using concatenation we can write the definition of a regular palindrome A in the form where Xk e {a, 1, 2,... 9} for k=1, 2, 3,... n, exce~t Xti:O Examples and Identification: The digits 1, 2,..., 9 are trivially palindromes. The only 2-digit palindromes are 11,22,33,... 99. Of course, palindromes are easy to identify by visual inspection. We see at once that 5493945 is a palindrome. In this study we will also refer to this type of palindromes as regular palindromes since we will later defme another type ofpalhidromes. As we have seen, palindromes are easily identitied by visual inspection, something we will have difficulties to do with, say prime numbers. Nevertheless, we need an algorithm to identifY palindromes because we can not use our visual inspection method on integers that occur in computer analysis of various sets of numbers. The 23 following routine, written in Ubasic, is built into various computer programs in this study