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A permutable prime, also known as anagrammatic prime, is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
which, in a given
base Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider) Base (stylized as BASE) is the third largest of Belgium Belgium ( nl, België ; french: Belgique ; german: Belgien ), officially the Kingdom of Belgium, ...

, can have its digits' positions switched through any
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes. In
base 10 The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...

, all the permutable primes with fewer than 49,081 digits are known :, , , , , , 17, 31, 37, 71, 73, 79, 97, 113,
131 131 may refer to: *131 (number) 131 (one hundred ndthirty-one) is the natural number following 130 (number), 130 and preceding 132 (number), 132. In mathematics 131 is a Sophie Germain prime, an irregular prime, the second 3-digit palindromic pr ...
,
199 Year 199 ( CXCIX) was a common year starting on Monday A common year starting on Monday is any non-leap year A leap year (also known as an intercalary year or year) is a that contains an additional day (or, in the case of a , a month) adde ...
, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031, ... Of the above, there are 16 unique permutation sets, with smallest elements :2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 199, 337, R19, R23, R317, R1031, ... Note R''n'' = $\tfrac$ is a
repunit In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreati ...
, a number consisting only of ''n'' ones (in
base 10 The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...

). Any
repunit prime In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreati ...
is a permutable prime with the above definition, but some definitions require at least two distinct digits. All permutable primes of two or more digits are composed from the digits 1, 3, 7, 9, because no prime number except 2 is even, and no prime number besides 5 is divisible by 5. It is provenA.W. Johnson, "Absolute primes," ''Mathematics Magazine'' 50 (1977), 100–103. that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7, 9. There is no ''n''-digit permutable prime for 3 < ''n'' < 6·10175 which is not a repunit. It is
conjecture In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
d that there are no non-repunit permutable primes other than those listed above. In base 2, only repunits can be permutable primes, because any 0 permuted to the ones place results in an even number. Therefore, the base 2 permutable primes are the
Mersenne prime A Mersenne prime is a prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a compo ...
s. The generalization can safely be made that for any
positional number system Positional notation (or place-value notation, or positional numeral system) denotes usually the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
, permutable primes with more than one digit can only have digits that are
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
with the
radix In a positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally, a positional system is a numeral system in which the contribution ...

of the number system. One-digit primes, meaning any prime below the radix, are always trivially permutable. In
base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally ...
, the smallest elements of the unique permutation sets of the permutable primes with fewer than 9,739 digits are known (using inverted two and three for ten and eleven, respectively) :2, 3, 5, 7, Ɛ, R2, 15, 57, 5Ɛ, R3, 117, 11Ɛ, 555Ɛ, R5, R17, R81, R91, R225, R255, R4ᘔ5, ... There is no ''n''-digit permutable prime in base 12 for 4 < ''n'' < 12144 which is not a repunit. It is conjectured that there are no non-repunit permutable primes in base 12 other than those listed above. In base 10 and base 12, every permutable prime is a repunit or a near-repdigit, that is, it is a permutation of the integer ''P''(''b'', ''n'', ''x'', ''y'') = ''xxxx''...''xxxy''''b'' (''n'' digits, in base ''b'') where ''x'' and ''y'' are digits which is coprime to ''b''. Besides, ''x'' and ''y'' must be also coprime (since if there is a prime ''p'' divides both ''x'' and ''y'', then ''p'' also divides the number), so if ''x'' = ''y'', then ''x'' = ''y'' = 1. (This is not true in all bases, but exceptions are rare and could be finite in any given base; the only exceptions below 109 in bases up to 20 are: 13911, 36A11, 24713, 78A13, 29E19 (M. Fiorentini, 2015).) Let ''P''(''b'', ''n'', ''x'', ''y'') be a permutable prime in base ''b'' and let ''p'' be a prime such that ''n'' ≥ ''p''. If ''b'' is a primitive root of ''p'', and ''p'' does not divide ''x'' or , ''x'' - ''y'', , then ''n'' is a multiple of ''p'' - 1. (Since ''b'' is a primitive root mod ''p'' and ''p'' does not divide , ''x'' − ''y'', , the ''p'' numbers ''xxxx''...''xxxy'', ''xxxx''...''xxyx'', ''xxxx''...''xyxx'', ..., ''xxxx''...''xyxx''...''xxxx'' (only the ''b''''p''−2 digit is ''y'', others are all ''x''), ''xxxx''...''yxxx''...''xxxx'' (only the ''b''''p''−1 digit is ''y'', others are all ''x''), ''xxxx''...''xxxx'' (the
repdigit In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit. Examples ...
with ''n'' ''x''s) mod ''p'' are all different. That is, one is 0, another is 1, another is 2, ..., the other is ''p'' − 1. Thus, since the first ''p'' − 1 numbers are all primes, the last number (the repdigit with ''n'' ''x''s) must be divisible by ''p''. Since ''p'' does not divide ''x'', so ''p'' must divide the repunit with ''n'' 1s. Since ''b'' is a primitive root mod ''p'', the multiplicative order of ''n'' mod ''p'' is ''p'' − 1. Thus, ''n'' must be divisible by ''p'' − 1) Thus, if ''b'' = 10, the digits coprime to 10 are . Since 10 is a primitive root mod 7, so if ''n'' ≥ 7, then either 7 divides ''x'' (in this case, ''x'' = 7, since ''x'' ∈ ) or , ''x'' − ''y'', (in this case, ''x'' = ''y'' = 1, since ''x'', ''y'' ∈ . That is, the prime is a repunit) or ''n'' is a multiple of 7 − 1 = 6. Similarly, since 10 is a primitive root mod 17, so if ''n'' ≥ 17, then either 17 divides ''x'' (not possible, since ''x'' ∈ ) or , ''x'' − ''y'', (in this case, ''x'' = ''y'' = 1, since ''x'', ''y'' ∈ . That is, the prime is a repunit) or ''n'' is a multiple of 17 − 1 = 16. Besides, 10 is also a primitive root mod 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, ..., so ''n'' ≥ 17 is very impossible (since for this primes ''p'', if ''n'' ≥ ''p'', then ''n'' is divisible by ''p'' − 1), and if 7 ≤ ''n'' < 17, then ''x'' = 7, or ''n'' is divisible by 6 (the only possible ''n'' is 12). If ''b'' = 12, the digits coprime to 12 are . Since 12 is a primitive root mod 5, so if ''n'' ≥ 5, then either 5 divides ''x'' (in this case, ''x'' = 5, since ''x'' ∈ ) or , ''x'' − ''y'', (in this case, either ''x'' = ''y'' = 1 (That is, the prime is a repunit) or ''x'' = 1, ''y'' = 11 or ''x'' = 11, ''y'' = 1, since ''x'', ''y'' ∈ .) or ''n'' is a multiple of 5 − 1 = 4. Similarly, since 12 is a primitive root mod 7, so if ''n'' ≥ 7, then either 7 divides ''x'' (in this case, ''x'' = 7, since ''x'' ∈ ) or , ''x'' − ''y'', (in this case, ''x'' = ''y'' = 1, since ''x'', ''y'' ∈ . That is, the prime is a repunit) or ''n'' is a multiple of 7 − 1 = 6. Similarly, since 12 is a primitive root mod 17, so if ''n'' ≥ 17, then either 17 divides ''x'' (not possible, since ''x'' ∈ ) or , ''x'' − ''y'', (in this case, ''x'' = ''y'' = 1, since ''x'', ''y'' ∈ . That is, the prime is a repunit) or ''n'' is a multiple of 17 − 1 = 16. Besides, 12 is also a primitive root mod 31, 41, 43, 53, 67, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, ..., so ''n'' ≥ 17 is very impossible (since for this primes ''p'', if ''n'' ≥ ''p'', then ''n'' is divisible by ''p'' − 1), and if 7 ≤ ''n'' < 17, then ''x'' = 7 (in this case, since 5 does not divide ''x'' or ''x'' − ''y'', so ''n'' must be divisible by 4) or ''n'' is divisible by 6 (the only possible ''n'' is 12).

# References

{{DEFAULTSORT:Permutable Prime Base-dependent integer sequences Classes of prime numbers Permutations