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Type of prime number
system. One-digit primes, meaning any prime below the radix, are always trivially permutable. In base 10, all the permutable primes with fewer than 49
Permutable_prime
Type of prime number
circular primes are the permutable primes, which are a subset of the circular primes (every permutable prime is also a circular prime, but not necessarily
Circular_prime
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no divisors other than 1 and itself
List_of_prime_numbers
Natural number
thus it is also a permutable prime. 37 is the median value for the second prime factor of an integer. 37 is the first irregular prime with irregularity
37_(number)
Type of number
10n−1, in order to match a decimal n-digit number with no leading 0. Permutable prime Sloane, N. J. A. (ed.). "Sequence A077390". The On-Line Encyclopedia
Truncatable_prime
Natural number
right-truncatable prime, because when the last digit (9) is removed, the remaining number (7) is still prime. Because 97 is prime, it is an emirp and a permutable prime
79_(number)
Class of prime numbers
every non-negative integer n {\displaystyle n} ). All non-palindromic permutable primes are emirps. It is not known whether there are infinitely many emirps
Emirp
Natural number
Where 73 and 37 are part of the sequence of permutable primes and emirps in base-ten, a Sheldon prime as defined as satisfying "mirror" and "product"
73_(number)
Prime number of the form 2^n – 1
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some
Mersenne_prime
Natural number
form 3 n − 1 {\displaystyle 3n-1} . In decimal, this prime is a primeval number and a permutable prime with 131 and 311. 113 is the denominator of 355/113
113_(number)
Natural number
square. 311 is a twin prime with 313, an irregular prime, an emirp, and a permutable prime with 113 and 131. It is an Eisenstein prime with no imaginary part
300_(number)
Natural number
can be written as the sum of two prime numbers that do not exceed p {\displaystyle p} . The twenty-third permutable prime in decimal R 19 {\displaystyle
23_(number)
Natural number
groups). In base ten, (17, 71) form the seventh permutation class of permutable primes. The sequence of residues (mod n) of a googol and googolplex, for
17_(number)
Prime such that p^2 divides 2^(p-1)-1
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem
Wieferich_prime
Natural number
20th prime number. Because both rearrangements of its digits (17 and 71) are prime numbers, 71 is an emirp and more generally a permutable prime. 71 is
71_(number)
Natural number
132. 131 is a Sophie Germain prime, an irregular prime, the second 3-digit palindromic prime, and also a permutable prime with 113 and 311. It can be expressed
131_(number)
Numbers that contain only the digit 1
exponent of the (N−1)th. The prime repunits are a trivial subset of the permutable primes, i.e., primes that remain prime after any permutation of their
Repunit
Natural number
e\approx {163 \over 3\cdot 4\cdot 5}\approx 2.7166\dots } 163 is a permutable prime in base 12, which it is written as 117, the permutations of its digits
163_(number)
Factorial prime Permutable prime Palindromic prime Cuban prime Lucky prime Ulam spiral Magic star Magic square Frénicle standard form Prime reciprocal
List of recreational number theory topics
List_of_recreational_number_theory_topics
Natural number
of the third and last trio of three-digit permutable primes in decimal: 199 + 919 + 991 2113 – Proth prime, centered square number 2120 – Fine number
2000_(number)
Prime number congruent to 1 mod 4
A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} . Pythagorean primes are exactly the odd prime numbers that are the sum
Pythagorean_prime
Type of prime number
In number theory, a Wilson prime is a prime number p {\displaystyle p} such that p 2 {\displaystyle p^{2}} divides ( p − 1 ) ! + 1 {\displaystyle (p-1)
Wilson_prime
group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect
Quasinormal_subgroup
Prime number of the form 2^u × 3^v + 1
In number theory, a Pierpont prime is a prime number of the form 2 u ⋅ 3 v + 1 {\displaystyle 2^{u}\cdot 3^{v}+1\,} for some nonnegative integers u and
Pierpont_prime
Natural number
intersection. 733 is a prime number, a balanced prime, a permutable prime, and an emirp. It is the sum of five consecutive primes (137 + 139 + 149 + 151
700_(number)
Prime number that is product of first n primes ± 1
mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes). Primality tests
Primorial_prime
Prime number of the form that allows fast modular reduction
In mathematics, a Solinas prime, or generalized Mersenne prime, is a prime number that has the form f ( 2 m ) {\displaystyle f(2^{m})} , where f ( x )
Solinas_prime
Type of prime number
A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers
Cuban_prime
reversed. A002113 Permutable primes 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, ... The numbers for which every permutation of digits is a prime. A003459 Harshad
List_of_integer_sequences
Type of natural number in recreational number theory
= 7×1Ɛ, and 135 = 5×31. Permutable prime Truncatable prime Chris Caldwell, The Prime Glossary: Primeval number at The Prime Pages Mike Keith, Integers
Primeval_number
Infinitely many prime numbers exist
statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his work Elements. There are at
Euclid's_theorem
Prime number of the form (2ᵖ+1)/3
theory, a Wagstaff prime is a prime number of the form 2 p + 1 3 {\displaystyle {{2^{p}+1} \over 3}} where p is an odd prime. Wagstaff primes are named after
Wagstaff_prime
Prime number one less or more than a factorial
factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for
Factorial_prime
Lévy–Steinitz theorem Antisymmetrizer Identical particles Levi-Civita symbol Permutable prime Bit-reversal permutation Claw-free permutation Heap's algorithm Permutation
List_of_permutation_topics
{\displaystyle K} whose order is relatively prime to that of H {\displaystyle H} . Clearly, every permutable subgroup of a finite group is semipermutable
Semipermutable_subgroup
Special type of prime number
In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem
Wolstenholme_prime
Positive integer of the form (2^(2^n))+1
If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. As of 2026[update]
Fermat_number
Result on density of prime numbers
that for any integer n > 3 {\displaystyle n>3} , there exists at least one prime number p {\displaystyle p} with n < p < 2 n − 2. {\displaystyle n<p<2n-2
Bertrand's_postulate
Number of form 2^(2^p-1)-1 with prime exponent
number that is prime is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime for a proof), a
Double_Mersenne_number
Mathematical concept
Cullen primes at The Prime Pages. The Prime Glossary: Cullen number at The Prime Pages. Chris Caldwell, The Top Twenty: Generalized Cullen at The Prime Pages
Cullen_number
Number of the form (n * 2^n) - 1
infinitely many Woodall primes? More unsolved problems in mathematics Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents
Woodall_number
Integer of the form 3 × 2^n – 1 for non-negative n
"10" followed by n 1s. The first few Thabit numbers that are prime (Thabit primes or 321 primes): 2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831
Thabit_number
Number sequence 3,0,2,3,2,5,5,7,10,...
0)\\8&2P(2)+3P(1)+2P(0)&P(2)-2P(1)+P(0)\end{array}}} The first fourteen prime Perrin numbers are In 1876 the sequence and its equation were initially
Perrin_number
Solution to x*x + y*y + z*z = 3xyz
to obtain a new Markov triple from an old one (x, y, z). First, one may permute the 3 numbers x,y,z, so in particular one can normalize the triples so
Markov_number
Number of the form x^y + y^x
Leyland numbers (so we have 1 < y ≤ x). A Leyland prime is a Leyland number that is prime. The first such primes are: 17, 593, 32993, 2097593, 8589935681, 59604644783353249
Leyland_number
American businessman and media personality (born 1978)
athletes, and business leaders, including U.S. President Donald Trump, Israeli Prime Minister Benjamin Netanyahu, and basketball player Kobe Bryant. In 2009
Patrick_Bet-David
Algebraic structure with addition, multiplication, and division
a prime field if it has no proper (i.e., strictly smaller) subfields. Any field F contains a prime field. If the characteristic of F is p (a prime number)
Field_(mathematics)
Method of random selection
number of legs are called "prime". A ghost leg can be constructed arbitrarily, but such a ghost leg is not necessarily prime. It can be proven that only
Ghost_leg
Solved prime-number problem
theorem) states that, for each n ≥ 2 {\displaystyle n\geq 2} , there is a prime p {\displaystyle p} such that n < p < 2 n {\displaystyle n<p<2n} . First
Proof_of_Bertrand's_postulate
Decimal representation of a number whose digits are periodic
prime is a proper prime if and only if it is a full reptend prime and congruent to 1 mod 10. If a prime p is both full reptend prime and safe prime,
Repeating_decimal
Discrete Fourier transform for prime sizes
Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution (the other algorithm for FFTs of prime sizes, Bluestein's algorithm
Rader's_FFT_algorithm
Algorithm for generating pseudo-randomized numbers
multiplier of a are equivalent to LCGs with a large prime modulus of abr−1 and a power-of-2 multiplier b. A permuted congruential generator begins with a power-of-2-modulus
Linear_congruential_generator
Natural number
92 is a composite number of the general form p2q, where q is a higher prime (23). It is the tenth of this form and the eighth of the form 22q. 92 has
92_(number)
Product of numbers from 1 to n
exponential growth. Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing
Factorial
Theorem on modular exponentiation
proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating
Euler's_theorem
American novelist
of Superior Achievement in a Young Adult Novel (Intentional Haunting, Permuted Press). Bible is the author of the Apex Trilogy, Z-Burbia and other series
Jake_Bible
Mathematical group
U 2 B R ′ D 2 R B ′ U 2 , {\displaystyle BR^{\prime }D^{2}RB^{\prime }U^{2}BR^{\prime }D^{2}RB^{\prime }U^{2},} (twist two corners) R U D B 2 U 2 B ′
Rubik's_Cube_group
Natural number
number following 222 and preceding 224. 223 is a lucky prime and a left-and-right-truncatable prime. Among the 720 permutations of the numbers from 1 to
223_(number)
Representation of symmetric groups
where two labellings are equivalent if one is obtained from the other by permuting the entries of each row. For each Young tableau T of shape λ let { T }
Specht_module
Mathematical theorem
inf f ′ ≤ f ′ ( c ) ≤ sup f ′ {\displaystyle \inf f^{\prime }\leq f^{\prime }(c)\leq \sup f^{\prime }} , the inequality above is a useful substitute. Moreover
Symmetry of second derivatives
Symmetry_of_second_derivatives
Complex-valued arithmetic function
Lejeune Dirichlet, who introduced these functions in his 1837 paper on primes in arithmetic progressions. They are a prominent example of the general
Dirichlet_character
Mathematical statement of uniqueness, except for an equivalent structure
the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer
Up_to
Mathematical problem
q is a prime or prime power. This follows from a construction that is based on a finite field GF(q), which only exist if q is a prime or prime power.
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Ordering of binary values, used for positioning and error correction
"cyclic binary code", "cyclic progression code", "cyclic permuting binary" or "cyclic permuted binary" (CPB). The Gray code is sometimes misattributed
Gray_code
Basic elements of language
independent carrier of meaning in a lexicon; and syntactically, as the smallest permutable and substitutable unit of a sentence. In some languages, these different
Word
Sporadic simple group
append 3 new points and let the automorphisms in PΓL(3,4) but not in M21 permute these new points. An S(3,6,22) system W22 is formed by appending just one
Mathieu_group_M24
Theorem on the orders of subgroups
to show that there are infinitely many primes: suppose there were a largest prime p {\displaystyle p} . Any prime divisor q {\displaystyle q} of the Mersenne
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
Subgroup invariant under conjugation
Descendant subgroup Quasinormal subgroup Seminormal subgroup Conjugate permutable subgroup Modular subgroup Pronormal subgroup Paranormal subgroup Polynormal
Normal_subgroup
Cryptographic hash function
hash function based on Daniel J. Bernstein's ChaCha stream cipher, but a permuted copy of the input block, XORed with round constants, is added before each
BLAKE_(hash_function)
Unsolved problem in mathematics
the cyclic group Z/nZ for any positive integer n. To do this, choose a prime p such that p ≡ 1 (mod n); this is possible by Dirichlet's theorem. Let
Inverse_Galois_problem
American sitcom (1987–1997)
ISBN 1629331899 Married... with Children vs. the World, by Richard Gurman, Permuted Press, April 2024, ISBN 9781637588314 Married... with Children was adapted
Married..._with_Children
Measure of dependence between two variables
) p ′ ′ ( w , y ) {\displaystyle p(x,y)\approx \sum _{w}p^{\prime }(x,w)p^{\prime \prime }(w,y)} Alternately, one might be interested in knowing how much
Mutual_information
Algorithm for shuffling a finite sequence
random integer such that i ≤ j ≤ n − 1 exchange a[i] and a[j] This example permutes the letters from A to H using Fisher and Yates' original method, starting
Fisher–Yates_shuffle
Construction in group theory
Gal(K / k), where k is the prime field for K; this is the fundamental theorem of projective geometry. Thus for K a prime field (Fp or Q), PGL = PΓL,
Projective_linear_group
Polynomial that permutes a ring
when s divides q − 1, and r > 1 is relatively prime (coprime) to q − 1, then xr(g(xs))(q - 1)/s permutes GF(q). Only a few other specific classes of permutation
Permutation_polynomial
Permutation that reverses binary numbers
concatenation method) from the steps that use the results of this calculation to permute the data (for instance, by scanning the data indexes in order and performing
Bit-reversal_permutation
U.S. Army installation in North Carolina
Regiment: The 504th Parachute Infantry Regiment in Sicily and Salerno. Permuted Press LLC, 2022. Wikimedia Commons has media related to Fort Bragg (North
Fort_Bragg
Mathematical metric in geometry
A homography on the larger space may have K as an invariant set as it permutes the points of the space. Such a homography induces one on P(R), and since
Cayley–Klein_metric
Product of prime numbers, plus one
prime numbers). They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime
Euclid_number
Alternatively, a group G is called an Iwasawa group when every subgroup of G is permutable in G (Ballester-Bolinches, Esteban-Romero & Asaad 2010, pp. 24–25). The
Iwasawa_group
that a p ≡ a ( mod p ) {\displaystyle a^{p}\equiv a{\pmod {p}}} for every prime number p and every integer a (see modular arithmetic). Some of the proofs
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
German cipher machine during World War II
model has 4 rotors (lines 1 through 4) and the reflector (line R) also permutes (garbles) letters. The Enigma family included multiple designs. The earliest
Enigma_machine
Realistic artificially generated media
2021. Retrieved 15 April 2021. Cohen, Ariel; Rimon, Inbal; Aflalo, Eran; Permuter, Haim H. (June 2022). "A study on data augmentation in voice anti-spoofing"
Deepfake
Theorem in group theory
rise to S 3 {\displaystyle S_{3}} is the non-trivial automorphism that permutes the two non-identity elements of C 3 {\displaystyle C_{3}} . Furthermore
Schur–Zassenhaus_theorem
Graph defined from a mathematical group
multiplication maps, for example group automorphisms of G {\displaystyle G} which permute S {\displaystyle S} . The Cayley graph Γ ( G , S ) {\displaystyle \Gamma
Cayley_graph
Value indicating the relation of a proposition to truth
"for every number there is a prime larger than it" is the set of all programs that take as input a number n, and output a prime larger than n. In category
Truth_value
Transformations induced by a mathematical group
{\displaystyle S_{n}} acts on any set with n {\displaystyle n} elements by permuting the elements of the set. Although the group of all permutations of a set
Group_action
Natural number
that have at least one sink vertex, with no outgoing edges, 185 ways of permuting the squares of a 2 × 4 {\displaystyle 2\times 4} grid of squares in such
185_(number)
Number of orderings allowing ties
product of distinct prime numbers; 30 is squarefree, but 20 is not, because its prime factorization 2 · 2 · 5 repeats the prime 2. For squarefree numbers
Ordered_Bell_number
Linear map that preserves areas
Some insight into logarithms comes through hyperbolic sectors that are permuted by squeeze mappings while preserving their area. The area of a hyperbolic
Squeeze_mapping
Sequence that reads the same forwards and backwards
written in 1964, consists of twenty sections, called "moments", which may be permuted in several different ways, including retrograde presentation, and two versions
Palindrome
Sporadic simple group
,t^{X}} . PSL 2 ( 11 ) {\displaystyle \operatorname {PSL} _{2}(11)} permutes these involutions under the exceptional 11-point representation, so they
Janko_group_J1
Model of quantum computing
\cdots ,x_{n}\rangle )=|f(x_{1},x_{2},\cdots ,x_{n})\rangle .} Note that Wf permutes the computational basis states. Of particular importance is the controlled
Quantum_circuit
Theorem in mathematics
{\displaystyle \|f(x)-f(x^{\prime })-x+x^{\prime }\|\leq \|x-x^{\prime }\|\,\sup _{0\leq t\leq 1}\|f^{\prime }(x+t(x^{\prime }-x))-I\|.} Now choose δ >
Inverse_function_theorem
Concept in abstract algebra
CW complex of dimension n with a free action of a discrete group G that permutes the cells, then cd Z ( G ) ≤ n {\displaystyle \operatorname {cd} _{\mathbb
Cohomological_dimension
Polynomial-time algorithm for the assignment problem
that the total cost of assignment is minimum. This can be expressed as permuting the rows of a cost matrix C to minimize the trace of a matrix, min P Tr
Hungarian_algorithm
Number that is the product of factorials
permuting the two children of the upper left vertex, 2 ways of permuting the two children of the upper middle vertex, and 5! = 120 ways of permuting the
Jordan–Pólya_number
Generalization of the binomial theorem to other polynomials
{\binom {n}{k_{1},\ldots ,k_{m}}}} is also the number of distinct ways to permute a multiset of n elements, where ki is the multiplicity of each of the ith
Multinomial_theorem
Type of group in mathematics
inversion, and the symmetric group Sn acts on both {±1}n and T × {1} by permuting factors. The elements of the Weyl group are represented by matrices in
Orthogonal_group
Group whose operation is a composition of braids
by a string to the boundary of the disk; each mapping homomorphism that permutes two of the punctures can then be seen to be a homotopy of the strings,
Braid_group
PERMUTABLE PRIME
PERMUTABLE PRIME
Boy/Male
Hindu, Indian
Chief; Prime
Boy/Male
Indian, Tamil
Important; Prime
Girl/Female
Afghan, Arabic, German, Hindu, Indian, Iranian, Muslim, Parsi, Sindhi, Turkish
Spring Season; Prime of Life; Bloom of Youth
Surname or Lastname
English
English : unexplained.Serbian : unexplained.
Surname or Lastname
English
English : variant of Prime, or from an Old English personal name Preng.
Surname or Lastname
German
German : of uncertain origin; possibly from the Latin personal name Primus (‘the first’), borne by several saints; or one composed with a Germanic word meaning ‘to prick or stab’; or from a personal name of Slavic origin Primm, from prēmu ‘right’.French : from a personal name (from Latin Primus).French : nickname from Old French prim ‘first’, possibly given to the eldest child in a family, or alternatively a nickname from Old French and Occitan prim ‘shrewd’, ‘clever’, ‘artful’, ‘sly’.Dutch : variant of Priem.English : variant of Prime.Some of the Prim families in VT descend from a Simon Laval dit Printemps, who was known in English-speaking areas as Seymour Prim.
Girl/Female
Persian American English Greek
Child of light. Famous Bearer: Margaret Thatcher, former Prime Minister of the United Kingdom.
Girl/Female
Persian American
Child of light. Famous Bearer: Margaret Thatcher, former Prime Minister of the United Kingdom.
Surname or Lastname
English
English : from a Middle English personal name or nickname. The personal name existed in Old English, and is probably derived from Old English prim ‘early morning’ (from Latin primus ‘first’, used as the name of one of the canonical hours). The surname may be derived from this word as a Middle English nickname in the sense ‘fine’, ‘excellent’.French : feminine form of Prim 3.Dutch : variant of Priem.Probably an Americanized spelling of German Preim, a topographic name (of Slavic origin), perhaps from a river near Hannover; or of Preime, a variant of Primus.
Boy/Male
Arabic, Muslim
Prime Minister
Boy/Male
Arabic, Muslim, Pashtun
Prime Chief
Boy/Male
Arabic, Muslim, Sindhi
Start; Forefront; Dawn; Bosom; Prime; The Highest Part; Heart
Girl/Female
English American Israeli
The precious metal.. Late prime minister of Israel Golda Meir.
Boy/Male
Muslim
Prime minister
Girl/Female
Arabic, Muslim
Best Selected; The Best Part; Elite; Top; Prime; Flower
Girl/Female
Persian American
Child of light. Famous Bearer: Margaret Thatcher, former Prime Minister of the United Kingdom.
Boy/Male
Indian
Prime
Girl/Female
German, Hebrew, Hindu, Indian, Kannada, Sanskrit
Adornment; Jewel; The First; Primeval; Daughter of Earth; My Ornament; My Witness; Ornament
Boy/Male
Spanish
Born first.
Girl/Female
Australian, Christian, Danish, Greek, Latin, Swedish
Prime of Life; Youth; Goddess of Youth and Cup-bearer to the Gods; Granddaughter of Zeus and Hera
PERMUTABLE PRIME
PERMUTABLE PRIME
PERMUTABLE PRIME
PERMUTABLE PRIME
PERMUTABLE PRIME
PERMUTABLE PRIME
PERMUTABLE PRIME
adv.
In a primeval manner; in or from the earliest times; originally.
n.
Very durable; lasting; continuing long.
a.
Freely permeable by radiant heat.
n.
The quality or state of being permeable.
a.
Primeval.
a.
Permanent; durable.
adv.
In a prime manner; excellently.
a.
Admitting passage; capable of being penetrated by another body or substance; permeable; as, a pervious soil.
a.
To prepare; to make ready; to instruct beforehand; to post; to coach; as, to prime a witness; the boys are primed for mischief.
adv.
In a permeable manner.
v. i.
To last or endure for a long time; to be perdurable or lasting.
a.
Belonging to the first ages; pristine; original; primitive; primary; as, the primeval innocence of man.
n.
One who, or that which, primes
a.
Capable of being permeated, or passed through; yielding passage; passable; penetrable; -- used especially of substances which allow the passage of fluids; as, wood is permeable to oil; glass is permeable to light.
n.
A kind of type, of which there are two species; one, called long primer, intermediate in size between bourgeois and small pica [see Long primer]; the other, called great primer, larger than pica.
n.
Full of pores; having interstices in the skin or in the substance of the body; having spiracles or passages for fluids; permeable by liquids; as, a porous skin; porous wood.
a.
To mark with a prime mark.
a.
Capable of being permuted; exchangeable.
a.
Not permeable; not permitting passage, as of a fluid. through its substance; impervious; impenetrable; as, India rubber is impermeable to water and to air.
n.
The quality or state of being prime, or excellent.