Is based 10 used for calculations?

Hi guys, I had a quick look at the wiki but could not find anything on the following.

I am just reading a book on "mathematical cranks" by Underwood Dudley.

He writes about a guy who found out that using a duodecimal system (base 12), all mersenne primes greater than 7 end in either "27" or "x7". (He is using symbols resembling to x and e for expanding from decimal to duodecimal, so 'x' stands for 10 and 'e' for 11 in his duodecimal system).

So I got the idea that if this were really so, I would expect the mersenne prime search to work with base 12, filter all these numbers out and only test these..? Or are there other methods for spotting using base 10?

Hope the question is not too st00pid :) It just seemed to me that this passage in the book is implying that there is no such pattern using base 10 and I don't have the time to read up on the matter.

[wblipp]

There is a standard notation for number systems a little larger than 10 of using A, B, etc. for the additional digits. You see this most often in hexadecimal notation. But we wouldn't expect a crank to know that - that's one of the signs of a crank.

What the crank has found is a simple consequence of the fact that after the first few, all primes are 6k+1 or 6k+5. To see why this is so, calculate the sequence 2^n mod 144. It repeats at length 6 (64, 128, 112, 80, 16, 32, 64, ...). Only the cases corresponding to 128 and 32 can be prime (6k+1 and 6k+5), and these correspond to his two endings.

[wblipp]

Originally Posted by Googol

It just seemed to me that this passage in the book is implying that there is no such pattern using base 10

The base 10 pattern has 8 possible endings, corresponding to the fact that after the first few, all primes are 20k+1, 20k+3, 20k+7, 20k+9, 20k+11, 20k+13, 20k+17, or 20k+19. The sequence 2^n mod 100 repeats in a cycle of length 20. The possible endings are 51, 07, 27, 11, 47, 91, 71, and 87 respectively.

William