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

• And same for other digits

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

See if you can find more!

Note by Zakir Husain
4 months 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 $10$ and $1000$ (to exclude single-digit primes), here's what it found:

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

These are interesting in themselves, but what startled me was the fact that when I increased my search interval to $10,000$, the program couldn't find any new ones. And again with $100,000$ as the maximum. Currently it has checked all primes up to $1,000,000$ (which took $5$ minutes!) and still couldn't find any other than the $18$ listed above. So unless my code is dramatically flawed, it seems to suggest that there are none of these primes with more than $3$ 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.

- 4 months ago

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.

- 4 months ago

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.

- 4 months ago

Awesome! Thanks Thomas!

- 4 months ago

Thanks for you efforts!

- 4 months ago

Another one in 3 digits: (113)

- 4 months ago

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.

- 4 months ago