# Category: Classes of prime numbers

Factorial prime
A 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 n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are (se
Quartan prime
In mathematics, a quartan prime is a prime number of the form x4 + y4 where x and y are positive integers. The odd quartan primes are of the form 16n + 1. For example, 17 is the smallest odd quartan p
Williams number
In number theory, a Williams number base b is a natural number of the form for integers b ≥ 2 and n ≥ 1. The Williams numbers base 2 are exactly the Mersenne numbers.
Woodall prime
No description available.
Balanced prime
In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it alge
Good prime
A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies t
Wieferich prime
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, which states that every odd prime p divides 2p
Full reptend prime
In number theory, a full reptend prime, full repetend prime, proper prime or long prime in base b is an odd prime number p such that the Fermat quotient (where p does not divide b) gives a cyclic numb
Palindromic prime
In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conven
Ramanujan prime
In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.
Delicate prime
A delicate prime, digitally delicate prime, or weakly prime number is a prime number where, under a given radix but generally decimal, replacing any one of its digits with any other digit always resul
Emirp
An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. The term reversibl
Safe and Sophie Germain primes
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 11 is a Sophie Germa
Cuban prime
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 x and y.
Chen prime
A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2p + 2 therefore satisfies Chen's theorem. The Chen primes are
Truncatable prime
In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For
Fibonacci prime
A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime. The first Fibonacci primes are (sequence in the OEIS): 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 29
Newman–Shanks–Williams prime
In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number p which can be written in the form NSW primes were first described by , Daniel Shanks and Hugh C. Williams in 1981 during t
Permutable prime
A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Riche
Long prime
No description available.
Pillai prime
In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a mul
Lucky prime
No description available.
Prime triplet
In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4,
Repunit prime
No description available.
Right-truncatable prime
No description available.
Strong prime
In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory.
Left-truncatable prime
No description available.
Wagstaff prime
In number theory, a Wagstaff prime is a prime number of the form where p is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr.; the prime pages credit François Mora
Higgs prime
A Higgs prime, named after Denis Higgs, is a prime number with a totient (one less than the prime) that evenly divides the square of the product of the smaller Higgs primes. (This can be generalized t
Pierpont prime
In number theory, a Pierpont prime is a prime number of the form for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. They are named after the math
Minimal prime (recreational mathematics)
In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal prim
Genocchi prime
No description available.
Sophie Germain prime
No description available.
Twin prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a
List of prime numbers
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an in
Cousin prime
In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by
Integer sequence prime
In mathematics, an integer sequence prime is a prime number found as a member of an integer sequence. For example, the 8th Delannoy number, 265729, is prime. A challenge in empirical mathematics is to
Supersingular prime (moonshine theory)
In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group M, which is the largest sporadic simple group. There are precisely f
Cullen prime
No description available.
Fortunate prime
No description available.
Pythagorean prime
A Pythagorean prime is a prime number of the form . Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squar
Supersingular prime (algebraic number theory)
In algebraic number theory, a supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve E is defined over the rational numbers, then a pr
Strobogrammatic number
A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. In other words, the numeral looks the same right-side up and upside
Happy prime
No description available.
Motzkin prime
No description available.
Mersenne prime
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 integer n. They are named after Marin Mersenne,
Gaussian prime
No description available.
Safe prime
No description available.
Wolstenholme's theorem
In mathematics, Wolstenholme's theorem states that for a prime number , the congruence holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one m
Regular prime
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility
Proth prime
A Proth number is a natural number N of the form where k and n are positive integers, k is odd and . A Proth prime is a Proth number that is prime. They are named after the French mathematician Franço
Wilson prime
In number theory, a Wilson prime is a prime number such that divides , where "" denotes the factorial function; compare this with Wilson's theorem, which states that every prime divides . Both are nam
Solinas prime
In mathematics, a Solinas prime, or generalized Mersenne prime, is a prime number that has the form , where is a low-degree polynomial with small integer coefficients. These primes allow fast modular
In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is th
Lucas prime
No description available.
Wolstenholme prime
In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17,
Irregular prime
No description available.
Dihedral prime
A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation (normally or upside
Sexy prime
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and 11 − 5 = 6. The term "sexy prime" i
Wall–Sun–Sun prime
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.
Cluster prime
In number theory, a cluster prime is a prime number p such that every even positive integer k ≤ p − 3 can be written as the difference between two prime numbers not exceeding p. For example, the numbe
Eisenstein prime
In mathematics, an Eisenstein prime is an Eisenstein integer that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units {±1, ±ω, ±ω2}, a + bω i
Highly cototient prime
No description available.
Super-prime
Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime number
Circular prime
A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a circular p
Woodall number
In number theory, a Woodall number (Wn) is any natural number of the form for some natural number n. The first few Woodall numbers are: 1, 7, 23, 63, 159, 383, 895, … (sequence in the OEIS).
Stern prime
A Stern prime, named for Moritz Abraham Stern, is a prime number that is not the sum of a smaller prime and twice the square of a non zero integer. That is, if for a prime q there is no smaller prime
Primorial prime
In 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 show that pn# − 1 is prime for n = 2
Right prime
No description available.
Mills prime
No description available.
Cullen number
In mathematics, a Cullen number is a member of the integer sequence (where is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbe
Double Mersenne prime
No description available.