# permutable prime

## Primary tabs

Synonym:
absolute prime
Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

### 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.