Numbers can be palindromes
Palindromes and Prime Numbers
(a very special relationship)
31.12.2010
All natural numerical values that can be written in a badic number system as a complete (even number of digits) palindromic number> 11 are divisible by (b + 1) [corresponds to 11] (see also divisibility rules). This is easy to prove because the alternating cross sum of a complete palindrome number always results in "zero". Prime numbers can therefore only be palindromic numbers> 11 if they are incomplete (odd number of digits) (including voluntary palindromic numbers).
A special form of these prime numbers are the Mersenne prime numbers, because in the dual system they only consist of the digits "1".
To determine peculiarities and rules, I wrote a program with which I examined the number range from 1 to 100000 for palindromic numbers in all meaningful badic number systems.
Results:
There are numbers that are not written as a palindromic number> 11 in any badic number system.
Here is the list of these numbers, all of which are prime:
Strictly nonpalindromic numbers
All numbers that are written as incomplete palindromic numbers in badic number systems are prime numbers or the squares of prime numbers. In the number range up to 100000 this applies to all numerical values, with exception the number (3 ^ 2) = 9. The 9 in the dual system is 1001 and thus a complete palindrome number.
Among the prime numbers that cannot be written as palindromic numbers in any badic number system (conditions mentioned above), there are also the following prime number twins:
Nonpalindromic prime twins
The number 83160 becomes in 132 badic number systems written as a palindrome number and is therefore the Palindromic number in the range up to 100,000.
A list shows how often numerical values are written as palindromic numbers in badic number systems:
Frequencies of the multiple palindromic numbers
The complete list of all numbers up to 100,000 is available for download here:
(Attention  the packed text file is approx. 3 MB in size!)
Full text of the analysis

The further evaluations in the number range up to 10 million resulted in:
(16.12.2006)
evaluation
The prime numbers with the most palindromes in badic number systems have 21 bases in which they are a palindrome.
These four numbers are:
5654881 9168161 8996401 9189181
It is noticeable that many multiple palindromic primes end with the digit 1. This is also the case with the notation in many other badic number systems!
The largest gap between two nonpalindromic primes is 146 primes.
This area is listed here:
gap
The English term for the nonpalindromic numbers is:
Strictly nonpalindromic numbers in German Strictly nonpalindromic number.
Read more about this on Wikipedia.
Evaluations up to 50 million:
(25.12.2006)
Results up to 50 000 000
The prime number 41081041 is the smallest number that is a palindrome in 32 bases. In the spelling of all badic number systems, up to the hexadecimal system, this prime number ends with a 1.

Evaluations up to 1 billion:
(31.12.2006)
Results up to 1 billion
Tuple (related sequences of "strictly nonpalindromic primes" without palindromic primes in between)
(14.01.2007)
Tuple
Last change: December 31, 2010
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