Amazing primes!

I have got some interesting numbers with amazing properties, I named them 'every permutation primes'.

  • Suppose there is a prime \(\overline{AB}\) where A and B are digits then \(\overline{BA}\) must also be a prime if that number is a combination prime.

  • For a 3 digit prime number ABC\overline{ABC}, every possible combination of A, B and C have to be a prime (i.e.ACB,BAC,BCA,CAB,CBA\overline{ACB}, \overline{BAC}, \overline{BCA}, \overline{CAB}, \overline{CBA})

  • And same for other digits

For 2 digits I have founded some: 11,13,17,31,37,71,7311, 13, 17, 31, 37, 71, 73

See if you can find more!

Note by Zakir Husain
1 month, 1 week ago

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Hi Zakir. This is a really interesting concept! I just had a couple of thoughts.

First, might I suggest the name be "permutation primes" instead? It would just be a little more accurate since we have to check each permutation (not combination) of the numbers in the prime.

I wrote a Python program which runs through the prime numbers and checks to see which meet the criteria you described. Between 1010 and 10001000 (to exclude single-digit primes), here's what it found:

11,13,17,31,37,71,73,79,97,113,131,199,311,337,373,733,919,99111, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991

These are interesting in themselves, but what startled me was the fact that when I increased my search interval to 10,00010,000, the program couldn't find any new ones. And again with 100,000100,000 as the maximum. Currently it has checked all primes up to 1,000,0001,000,000 (which took 55 minutes!) and still couldn't find any other than the 1818 listed above. So unless my code is dramatically flawed, it seems to suggest that there are none of these primes with more than 33 digits. If anyone would like the (potentially confusing) code, please let me know.

I have NO idea why this is, but it's really cool!

Thanks for sharing! I love explorations like these.

@Zakir Husain @Thomas Raffill

David Stiff - 1 month ago

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The number of permutations grows with number of digits much faster than the number of primes, which is why I said it would get harder and harder to find these. So I would not be surprised if 991 is the highest one, but it might take a lot of work to give a real proof.

@David Stiff @Zakir Husain

Thomas Raffill - 1 month ago

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Actually I just looked up "repunit primes" and found some higher examples: the 19-digit repunit 1111111111111111111 and the 23-digit repunit 11111111111111111111111 are prime, and so are the 317-digit repunit and the 1031-digit repunit (too long to post here). There are also three much larger repunits (49081 digits, 109297 digits and 270343 digits) that are suspected but not yet proven to be prime, and there's a conjecture that there are infinitely many repunit primes. (My source for all this is primes.utm.edu.)

I also found that the concept already existed, it is called "permutable prime" or "anagrammic prime," and there is a wikipedia article on it https://en.wikipedia.org/wiki/Permutable_prime summarizing the known results about them. It says there is a conjecture that 991 is the largest non-repunit example.

@David Stiff @Zakir Husain

Thomas Raffill - 1 month ago

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@Thomas Raffill Awesome! Thanks Thomas!

David Stiff - 1 month ago

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Thanks for you efforts!

Zakir Husain - 1 month ago

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Another one in 3 digits: (113)

Zakir Husain - 1 month, 1 week ago

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Is there a maximum such number, or are there infinitely many? I haven't yet made a serious effort to search for these or prove any results, but it looks like it gets harder and harder to find them as the number of digits increases.

Thomas Raffill - 1 month, 1 week ago

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