decimal-binary prime pairs

ixfd64 · 2003-10-15, 23:31

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 (2² - 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 (2¹⁹ - 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.

2003-10-15, 23:31	#1
ixfd64 Bemusing Prompter "Danny" Dec 2002 California 2,467 Posts	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 (2² - 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 (2¹⁹ - 1) Probably no-one knows whether there exists an infinite amount of such pairs. Last fiddled with by ixfd64 on 2003-10-15 at 23:32

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2003-10-16, 05:00	#2
Fusion_power Aug 2003 Snicker, AL 1700₈ Posts	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

2003-10-16, 13:40	#3
I_like_tomatoes Oct 2003 Canada 47 Posts	It's interesting that both 3, (whose binary equivalent is 101), and the decimal number 101 are prime.