## Wednesday, October 15, 2014

### Interesting Numbers - Political Primes

Centrist, Democratic and Republican Primes
Tascino "Cino" Hilliard developed the idea of Centrist, Democratic and Republican primes.  They are listed in the Online Encyclopedia of Integers Sequences as sequences A125525, A125523, and A125524 and were added to the collection in 2007.  (see http://www.oeis.org).  With mid-term elections coming up I thought I would address these (even though they won’t effect the elections).
If you take a prime number and divide it in half (left and right halves, ignoring the middle digit if it has an odd number of digits), and you find that the left half of the number is prime, and the right half is not prime – then we call that Democratic prime.  Example: 29 is a prime number.  When we separate it into the left and right halves, we get 2 and 9.  The 2 on the left is a prime number, but the 9 on the right is not a prime number, so 29 fits the definition of a Democratic prime.
If you take a prime number and divide it in half (left and right halves, ignoring the middle digit if it has an odd number of digits), and you find that the left half of the number is not prime, but the right half is prime – then we call that Republican prime.  Example: 13 is a prime number.  When I separate the left and right halves I get 1 and 3.  1 (the number on the left) is not prime, but 3 (the number on the right) is prime.  So 13 fits the definition of a Republican prime.
If you take a prime number and divide it in half (left and right halves, ignoring the middle digit if it has an odd number of digits), and you find that the left half and the right half of the number are both prime, – then we call that Centrist prime.  Example: 23 is a prime number.  When I separate the left and right halves I get 2 and 3.  Since both 2 and 3 are prime numbers, 23 is considered a Centrist prime.
As I mentioned before, 13 is a Republican prime.  Well, let me be more specific.  13 is a Republican prime when it is written in base 10 (or decimal) as it is here.  If I write 13 as a base 2 (also known as binary).  13 = 11012  (The 2 written in subscript indicates that the number 1101 is written in base 2)  If I separate the left half and the right half I get 112 and 012.  112 = 3 in decimal, and 012 = 1 in decimal.  3 is a prime number and 1 is not a prime number.  So when 13 is written in base 2 it changes from a Republican prime to a Democratic prime.
When 13 is written in base 5 it becomes 235.  So when I separate them into the left and right halves I get 25 and 35.  Both of these are prime, so 13 is a Centrist prime when written in base 5.
So it seams that that Centrist, Democratic and Republican primes can change depending on what base they are written in.
As with many mathematicians, the next thing that came to my mind was “What is the biggest political prime that I could find?”
666 * 1046,396 + 7.  Written out this number is 666 followed by 46, 394 zeros with a 7 on the end.  This number is a Republican prime, a Beastly prime, and a Naughty prime.
If I could time-travel, I could go even larger.  But I will wait until later to explain.

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David