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# permutable prime

Given the base $b$ representation of a prime number $p$ as $d_{k},\ldots,d_{1}$ with

$p=\sum_{{i=1}}^{k}d_{i}b^{{i-1}},$ |

if each possible permutation of the digits still represents a prime number in that base, then $p$ is said to be a *permutable prime*. For example, in base 10, the prime 337 is a permutable prime since 373 and 733 are also prime. The known base 10 permutable primes are listed in A003459 of Sloane’s OEIS.

If we define $\pi_{P}(n)$ to count how many permutable primes there are below $n$, it is obvious that $\pi_{P}(b-1)=\pi(b-1)$, where $\pi(n)$ is the standard prime counting function.

When $2|b$, a search for permutable primes can safely exclude any primes whose base $b$ representation includes digits that are individually even. In a trivial sense, all repunit primes are also permutable primes. This means that in binary, the only permutable primes are repunits (that is, the Mersenne primes). Richert proved in 1951 that in the range $991<p<10^{{175}}$ the only base 10 permutable primes are repunit primes; it is conjectured that this is also true above that range.

# References

- 1 H. E. Richert, ”On permutable primtall,” Unsolved Norsk Matematiske Tiddskrift, 33 (1951), 50 - 54.

## Mathematics Subject Classification

11A63*no label found*

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## Comments

## Terms: absolute vs. permutable

It could be argued that "absolute" is more valid since the earliest researchers used this term. Personally, I prefer "permutable" because it indicates that permutation is involved, while "absolute" would seem to indicate an absoluteness that transcends base, which in this case it clearly does not (suffice it to compare the permutables in binary and decimal).

## little something to chew on

In

Table[Length[Select[Prime[Range[168]], permutablePrimeQ[#, b] &]], {b, 2, 36}]

Out

{4, 4, 8, 9, 10, 20, 13, 22, 22, 46, 17, 46, 27, 31, 26, 37, 23, 64, 37, 55, \

33, 61, 33, 77, 48, 68, 44, 109, 34, 100, 59, 72, 72, 92, 49}

Prime pi 1000 is 168. So, from the bases Mathematica can handle natively, base 29 has the most permutable primes from among the first 168 primes. The definition of the Boolean function permutablePrimeQ "is left as an exercise for the Reader."

## Re: little something to chew on

Obviously p < b will be permutable in a trivial sense, i.e., there's not much permutation you can do on just one digit. So then larger bases have more permutable primes in the beginning. A fairer comparison would be to look for permutable primes in the range b^2 < p < b^3.

## Re: little something to chew on

Mm, yes, I agree. Good points. Thanks.