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B Primes

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.

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Categories: Math
  1. Phynnboi
    February 9, 2009 at 2:27 pm

    Hmm…. Neither 0 nor 1 is a prime, so they can’t be B-primes. Two isn’t a B-prime, since removing the first leading digit from 10 makes 0, which is neither prime nor 1. One cannot remove “any” number of leading digits, since that includes removing all digits, making a non-number. Theorem 1 doesn’t make any sense since B-prime was defined only for base 2.

    I think I can see where you’re trying to go with this, but the definition and example sequence need to be cleaned up a bit. For instance, if we expand B prime to bases besides base 2; allow a number to be reduced a prime, 1, or 0; and mandate that the least significant digit cannot be removed; then theorems 1 and 2 are correct.

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