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B Primes

December 29, 2008 1 comment

B primes are primes where, in binary, any number of digits can be removed from the front and it is still prime or 1. This can work in any base, but I’m doing it in binary. In other bases, there are far fewer B primes. Also, it’s a lot easier in binary.

Here’s an example.

1011 is a B prime. 1011 in binary is 11 in base 10, which is prime.

If the first digit is removed, it becomes 011, which is 3 in base 10.
If the second digit is removed, it becomes 11, which is also 3 in base 10.
If the third digit is removed, it becomes 1, which is 1 in base 10.

I wrote a Java package to calculate B primes. It is currently being edited, and will become available soon.

The highest base ten B primes it was able to find is 17.

I believe I have found all the B primes up to 2^18.

New complete list of every B prime up to 2^20 (1048576):
0, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 43,
61, 67, 71, 83, 101, 107, 131, 139, 151, 157, 199,
211, 229, 257, 263, 269, 293, 317, 467, 523, 541,
613, 619, 643, 769, 829, 1031, 1061, 1091, 1163, 1181,
1223, 1637, 1667, 2053, 2131, 2179, 2311, 2341, 3079,
3109, 3229, 3271, 4099, 4133, 4139, 4157, 4253, 4637,
8209, 8221, 8263, 8293, 8461, 9283, 9829, 9859, 12829,
16421, 16427, 16451, 17027, 19463, 20483, 24593, 24677,
25667, 32771, 32779, 32797, 32839, 32869, 33037, 33829,
33931, 36901, 37021, 65537, 65539, 65543, 65579, 65687,
66179, 67589, 73757, 98573, 102437, 131101, 131143,
131203, 132103, 132709, 132739, 135211, 233509, 262147,
262151, 262187, 264323, 270407, 270437, 272003, 274973,
294923, 295973, 393287,

The highest B prime so far is 393287.

I have also proved several theorems about B primes:

1. All primes p are B primes in at least one base b where p > b.
2. All primes p are B primes in infinitely many bases b where b > p.
3. There are infinitely many B primes. (follows from 2)
4. If there are infinitely many primes in the form 2^p + 1, then there are infinitely many primes in base 2.

Categories: Math

The New Keyboard Layout Project (NKLP)

December 9, 2008 10 comments

12/9/08

Right now I am expanding my corpus. If anyone else has big blocks of text, like a bunch of stuff they typed on their computer, send it to me at MTGandP@gmail.com. Tell me what’s in it (like, emails, computer programs, business letters) so I don’t have to read it. (For confidentiality reasons, and for my convenience.)

12/11/08
I’m trying to get my corpus up to 10,000 pages, because I think that’s enough to have a really good variety of text. Right now I have about 3000.

12/12/08
I have about 11,000 pages in my corpus. However, I only have about 1000 pages of casual text, an I’d like about 3000. I’d also like about 1000 pages of news, and only have about 400. But I’m close to being done. (Collecting news is just so tedious, though.)

I just realized that Carpalx has a good corpus that’s free. It doesn’t have everything that I want, but it has a lot of books and programming code.

VvV from colemak.com wanted to make an evolutionary algorithm for non-latin letters. I don’t have any data for any languages other than English. But if I did, what languages could be done? I’ll look at the world’s most popular languages (from geography.about.com).

1. Mandarin Chinese – 882 million
There are far too many characters in Chinese to make a keyboard layout.
2. Spanish – 325 million
latin
3. English – 312-380 million
latin
4. Arabic – 206-422 million
This could work. http://en.wikipedia.org/wiki/Arabic_alphabet
5. Hindi – 181 million
This alphabet is also probably small enough. http://www.omniglot.com/writing/hindi.htm
6. Portuguese – 178 million
latin
7. Bengali – 173 million
It looks kind of big, but it should work. http://www.omniglot.com/writing/bengali.htm
8. Russian – 146 million
Yep. http://en.wikipedia.org/wiki/Russian_alphabet
9. Japanese – 128 million
Same deal as Chinese.
10. German – 96 million
latin

Any comments you have relating to the NKLP should be posted here.

Categories: Keyboards