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permutable prime
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(Definition)
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Given the base  representation of a prime number  as
 with
if each possible permutation of the digits still represents a prime number in that base, then  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  to count how many permutable primes there are below  , it is obvious that
 , where  is the standard prime counting function.
When  , a search for permutable primes can safely exclude any primes whose base  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
 the only base 10 permutable primes are repunit primes; it is conjectured that this is also true above that range.
- 1
- H. E. Richert, "On permutable primtall," Unsolved Norsk Matematiske Tiddskrift, 33 (1951), 50 - 54.
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"permutable prime" is owned by PrimeFan. [ owner history (1) ]
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(view preamble)
| Other names: |
absolute prime |
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Cross-references: range, Mersenne primes, repunits, binary, repunit primes, even, prime counting function, obvious, OEIS, represents, digits, permutation, prime number, representation, base
There is 1 reference to this entry.
This is version 2 of permutable prime, born on 2006-09-08, modified 2006-09-12.
Object id is 8328, canonical name is PermutablePrime.
Accessed 1184 times total.
Classification:
| AMS MSC: | 11A63 (Number theory :: Elementary number theory :: Radix representation; digital problems) |
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Pending Errata and Addenda
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