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 ixfd64 2003-10-15 23:31

decimal-binary prime pairs

I've been working on a type of prime number pairs I call "decimal-binary" primes. They are primes that consist of only 0's and 1's, like the binary system (repunits are OK). When the prime is used as a binary string, the result is a prime number, too. For example, 11 is prime, and in binary, 11 is 3, which is also prime. For obvious reasons, the number must end in 1.

Here are the pairs I found (I'm sure I missed a lot):

Tell me if I made any mistakes.

11, 3 (2[sup]2[/sup] - 1)
101, 5
101111, 47
1011001, 89
1100101, 101
10010101, 149
10100011, 163
10101101, 173
101001001, 337
101101111, 367
101111111, 383
110111011, 443
10100101001, 1321
11011011001, 1753
10101001010101, 10837
10101110100101, 11173
101011101010101011, 178859
1111111111111111111, 524287 (2[sup]19[/sup] - 1)

Probably no-one knows whether there exists an infinite amount of such pairs.

 Fusion_power 2003-10-16 05:00

If I am interpreting correctly, here is a verbal translation of your examples:

Eleven is prime. In binary 11 is 3 which is also prime.

One Hundred One is prime. In binary, 101 is 5 which is also prime.

One Hundred One Thousand One Hundred Eleven is prime and so is the binary equivalent of 47.

Given that the number of primes is proven infinite, a corollary would be that an infinite number of decimal/binary pairs should also exist.

Note the distribution curve if you graph the examples. Thats a kind of unique pattern.

Fusion

 I_like_tomatoes 2003-10-16 13:40

It's interesting that both 3, (whose binary equivalent is 101), and the decimal number 101 are prime.

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