# 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 b-adic 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 b-adic number systems.

Results:
There are numbers that are not written as a palindromic number> 11 in any b-adic number system.
Here is the list of these numbers, all of which are prime:
Strictly non-palindromic numbers

All numbers that are written as incomplete palindromic numbers in b-adic 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 b-adic number system (conditions mentioned above), there are also the following prime number twins:
Non-palindromic prime twins

The number 83160 becomes in 132 b-adic 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 b-adic 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

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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 b-adic 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 b-adic number systems!

The largest gap between two non-palindromic primes is 146 primes.
This area is listed here:
gap

The English term for the non-palindromic numbers is:
Strictly non-palindromic numbers in German Strictly non-palindromic number.

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 b-adic number systems, up to the hexadecimal system, this prime number ends with a 1.

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Evaluations up to 1 billion:
(31.12.2006)
Results up to 1 billion

Tuple (related sequences of "strictly non-palindromic primes" without palindromic primes in between)
(14.01.2007)
Tuple

Last change: December 31, 2010