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permutable prime

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

Mathematics Subject Classification

11A63 no label found

Comments

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).

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."

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.

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

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