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Number divisible only by 1 and itself
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that
Prime_number
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
Characterization of how many integers are prime
{\displaystyle \log _{e}(x)} . In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of prime numbers among the positive integers. It
Prime_number_theorem
The largest known prime number as of August 2026[update] is 2136,279,841 − 1, a number that has 41,024,320 digits when written in the decimal system. It
Largest_known_prime_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
Number representing illegal information
posted similar flags. An illegal prime is an illegal number which is also prime. One of the earliest illegal prime numbers was generated in March 2001
Illegal_number
Natural number
(two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and the only even prime number. Because it
2
Natural number
natural number following 72 and preceding 74. It is a prime number. 73 is a prime number, a twin prime (with 71), a Pierpont prime, and a star number. 73
73_(number)
Formula whose values are the prime numbers
In number theory, a formula for primes is a formula that outputs prime numbers. Such formulas for calculating primes do exist; however, they are computationally
Formula_for_primes
Prime numbers which differ by 6
In number theory, sexy primes are prime numbers that differ from another prime by 6. For example, the numbers 5 and 11 are a pair of sexy primes, because
Sexy_primes
Numbers with a certain property involving recursive summation
10-happy number. Paul Jobling discovered the prime in 2005. As of 2010[update], the largest known 10-happy prime is 242643801 − 1 (a Mersenne prime).[dubious
Happy_number
Difference between two successive prime numbers
is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive
Prime_gap
Numbers with many divisors
number n, the k given prime numbers pi must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by
Highly_composite_number
Natural number
(seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube. As an early prime number in the series of positive
7
Two numbers without shared prime factors
In number theory, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them
Coprime_integers
1019 is a prime number, a Sophie Germain prime, a safe prime, and a Chen prime. 1021 is a prime number, a Lucky prime, and a twin prime with 1019. 1025
1000_(number)
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
Function representing the number of primes less than or equal to a given number
is, the number of prime numbers less than x, plus half if x equals a prime. Of great interest in number theory is the growth rate of the prime-counting
Prime-counting_function
Natural number
57 (fifty-seven) is the natural number following 56 and preceding 58. 57 has prime factorization 3 ⋅ 19 {\displaystyle 3\cdot 19} , and is therefore a
57_(number)
Decomposition of a number into a product
composite number, or it is not, in which case it is a prime number. For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because
Integer_factorization
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
Numbers that contain only the digit 1
repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of May 2025, the largest known prime number
Repunit
Topics referred to by the same term
Prime number theory may refer to: Prime number Prime number theorem Number theory Fundamental theorem of arithmetic, which explains prime factorization
Prime_number_theory
Prime differing from another prime by two
_{e}(x)} . 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 (17, 19)
Twin_prime
Branch of number theory
either 1 or a prime number. However, it is strictly weaker. For example, −2 is not a prime number because it is negative, but it is a prime element. If
Algebraic_number_theory
Branch of pure mathematics
Number theory is a branch of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers
Number_theory
Palindromic prime number
Belphegor's prime is the palindromic prime number 1000000000000066600000000000001 (1030 + 666 × 1014 + 1), a number which reads the same both backwards
Belphegor's_prime
Natural number
(three) is a number, numeral and digit. It is the natural number following 2 and preceding 4, and is the smallest odd prime number and the only prime preceding
3
Sequence of numbers
Full reptend primes are italicised. † Unique primes are highlighted. A full reptend prime, full repetend prime, proper prime or long prime in base b is
Reciprocals_of_primes
Complex number whose real and imaginary parts are both integers
Gaussian prime. If z0 is a decomposed prime or the ramified prime 1 + i (that is, if its norm N(z0) is a prime number, which is either 2 or a prime congruent
Gaussian_integer
Algorithms to generate prime numbers
In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently. These are used in various applications
Generation_of_primes
Visualization of the prime numbers
problems in number theory such as Landau's problems. In particular, no quadratic polynomial has ever been proved to generate infinitely many primes, much less
Ulam_spiral
Used to count, measure, and label
even number with an integer is another even number; only the product of an odd number with an odd number is another odd number. A prime number, often
Number
Integer filtered out using a sieve similar to that of Eratosthenes
Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture
Lucky_number
Type of prime number
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
Regular_prime
Natural number
natural number following 58 and preceding 60. 59 is the 17th prime number, and 7th super-prime. It is also a good prime, a Higgs prime, an irregular prime, a
59_(number)
Natural number
natural number following 699 and preceding 701. It is a composite number and the sum of four consecutive primes (167 + 173 + 179 + 181). 701 is a prime number
700_(number)
Natural number
identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents
1
Natural number
natural number following 30 and preceding 32. 31 is a prime number, a twin prime (with 29), a super-prime, a member of an emirp prime pair, a sexy prime, and
31_(number)
Natural number
number following 66 and preceding 68. 67 is the 19th prime number, a Chen prime, an irregular prime, a lucky prime, a Heegner number, a super-prime,
67_(number)
Natural number
is a prime number, a Cuban prime, a Lucky prime a Chen prime, a centered triangular number, a centered hexagonal number, and a lazy caterer number (sequence
600_(number)
Natural number
the natural number following 22 and preceding 24. It is a prime number. Twenty-three is the ninth prime number and the smallest odd prime that is not
23_(number)
Natural number
smallest 12-digit prime number. 100,000,147,984 = 3162282, the smallest 12-digit square. 100,000,404,505 = smallest triangular number with 12 digits and
100,000,000,000
Natural number
consecutive primes (173 + 179 + 181 + 191 + 193). 918 = 2 × 33 × 17. It is a Harshad number. 919 is a prime number, a cuban prime, a prime index prime, a Chen
900_(number)
Natural number
number following 799 and preceding 801. It is the sum of four consecutive primes (193 + 197 + 199 + 211). It is a Harshad number, an Achilles number and
800_(number)
Natural number
symmetric number. 503 is a prime number, a safe prime, a Chen prime, an Eisenstein prime with no imaginary part, an index of a prime Lucas number, and an
500_(number)
Prime pair of the form (p, 2p+1)
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
Safe and Sophie Germain primes
Safe_and_Sophie_Germain_primes
Natural number
natural number following 42 and preceding 44. 43 is a prime number, and a twin prime of 41. 43 is the smallest prime that is not a Chen prime. 43 is also
43_(number)
Natural number
Keith number 148,149 = Kaprekar number 152,381 = unique prime in base 20 156,146 = Keith number 155,921 = smallest prime number being the only prime in an
100,000
Topics referred to by the same term
Largest prime number may refer to: Euclid's theorem, a statement that there are infinitely many prime numbers, and therefore no absolute largest Largest
Largest_prime_number
Natural number
consecutive primes in two different ways: 304 = 41+43+47+53+59+61 = 23+29+31+37+41+43+47+53. 305 = 5 × 61. It is the fifth hexagonal prism number which is
300_(number)
Type of number in mathematics
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
Palindromic_prime
Number of form 2^(2^p-1)-1 with prime exponent
Mersenne 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)
Double_Mersenne_number
Theorem on the number of primes in arithmetic sequences
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Natural number
(five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. Humans, and
5
Natural number
natural number following 18 and preceding 20. It is a prime number. 19 is the eighth prime number. 19 forms a twin prime pair with 17, a cousin prime pair
19_(number)
Natural number
natural number following 28 and preceding 30. It is a prime number. 29 is the number of days February has on a leap year. 29 is the tenth prime number. It
29_(number)
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin
List of Mersenne primes and perfect numbers
List_of_Mersenne_primes_and_perfect_numbers
Number equal to the sum of its proper divisors
{\displaystyle 2^{p}-1} with a prime p are prime; for example, 211 − 1 = 2047 = 23 × 89 is not a prime number. In fact, Mersenne primes are very rare: of the approximately
Perfect_number
Ancient algorithm for generating prime numbers
starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference
Sieve_of_Eratosthenes
Ideal in a ring which has properties similar to prime elements
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the
Prime_ideal
Prime fulfilling an inequality related to the prime-counting function
mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function. In 1919
Ramanujan_prime
Numbers obtained by adding the two previous ones
A Fibonacci prime is a Fibonacci number that is prime. The first few are: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... Fibonacci primes with thousands
Fibonacci_sequence
Natural number
It is a Ramanujan prime, a Pythagorean prime, a Chen prime, a super-prime, and an Irregular prime. It is also a Tetranacci number, a zero of Mertens
400_(number)
Natural number
natural number following 36 and preceding 38. 37 is the 12th prime number, and the 3rd isolated prime without a twin prime. 37 is a sexy prime, being 6
37_(number)
Natural number
Erdős–Nicolas number, triangular number, number of 5-cubes in a 9-cube, 2017 – sexy prime with 2011 2018 – Number of partitions of 60 into prime parts 2019
2000_(number)
Natural number
index prime because 13 is prime. The next prime number is 43, making both twin primes. It is a regular prime,a Ramanujan prime, a harmonic prime, a good
41_(number)
Natural number between 89 and 91
prime sextuplet, 113, is the 30th prime number. Since prime sextuplets are formed from prime members of lower order prime k-tuples, 90 is also a record maximal
90_(number)
Natural number
J3, and J1). 6 is the smallest integer which is not an exponent of a prime number, making it the smallest integer greater than 1 for which there does not
6
Natural number
natural number following 52 and preceding 54. 53 is a balanced prime, an isolated prime, the eighth Sophie Germain prime, and the ninth Eisenstein prime. It
53_(number)
Integers have unique prime factorizations
by some prime number.) Proposition 31 is proved directly by infinite descent. Any number either is prime or is measured by some prime number. — Euclid
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Natural number
007 : smallest prime number with 10 digits. 1,000,006,281 : smallest triangular number with 10 digits and the 44,721st triangular number. 1,000,014,129
1,000,000,000
Topics referred to by the same term
supersingular prime is a prime number satisfying one of the following concepts: Supersingular prime (algebraic number theory) Supersingular prime (moonshine
Supersingular_prime
Integer having a non-trivial divisor
Every composite number can be written as the product of two or more (not necessarily distinct) primes. For example, the composite number 299 can be written
Composite_number
Exploring properties of the integers with complex analysis
number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number
Analytic_number_theory
The tables contain the prime factorization of the natural numbers from 1 to 1000. When n is a prime number, the prime factorization is just n itself, written
Table_of_prime_factors
Product of two prime numbers
In number theory, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other
Semiprime
Natural number
thousand) is the natural number following 5999 and preceding 6001. 6028 – centered heptagonal number 6037 – super-prime, prime of the form 2p-1 6042 –
6000_(number)
Infinitely many prime numbers exist
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his
Euclid's_theorem
Prime number with a certain relationship to an elliptic curve
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
Supersingular prime (algebraic number theory)
Supersingular_prime_(algebraic_number_theory)
Study of subsets of integers and behavior under addition
sufficiently large odd number is the sum of three primes, and so every sufficiently large even integer is the sum of four primes. Hilbert proved that,
Additive_number_theory
Algebraic structure
is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p {\displaystyle
Finite_field
Type of number
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
Truncatable_prime
In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number which can be written in the form S 2 m + 1 = ( 1 + 2 ) 2 m + 1 + ( 1 − 2
Newman–Shanks–Williams_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
In number theory, Waring's prime number conjecture is a conjecture related to Vinogradov's theorem, named after the English mathematician Edward Waring
Waring's prime number conjecture
Waring's_prime_number_conjecture
Natural number
(forty-seven) is the natural number following 46 and preceding 48. It is a prime number. It is the adopted favorite number of Pomona College, a liberal
47_(number)
Algorithm for determining whether a number is prime
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike
Primality_test
Natural number
the natural number following 126 and preceding 128. It is also a prime number. As a Mersenne prime, 127 is related to the perfect number 8128. 127 is
127_(number)
Natural number
of all its divisors aside from one equals 53, which is the sixteenth prime number. There is no solution to the equation φ(x) = 34, making 34 a nontotient
34_(number)
Thought experiment of infinite sets
seat. Like the prime powers method, this solution leaves certain rooms empty. (In particular, any room with a number divisible by any prime other than 2
Hilbert's paradox of the Grand Hotel
Hilbert's_paradox_of_the_Grand_Hotel
Natural number
natural number following 78 and preceding 80. 79 is a Fortunate prime, a circular prime, a happy prime, a Higgs prime, a lucky prime, a regular prime, a cousin
79_(number)
term as prime minister, heading a coalition government after the BJP lost its outright majority in the Lok Sabha. Key No.: Incumbent number † Assassinated
List of prime ministers of India
List_of_prime_ministers_of_India
Count of the possible partitions of a set
a number N {\displaystyle N} is a squarefree positive integer, meaning that it is the product of some number n {\displaystyle n} of distinct prime numbers
Bell_number
Natural number
11 is a prime number, and a super-prime. 11 forms a twin prime pair with 13, and sexy prime pairs with both 5 and 17. 11 is also the first prime exponent
11_(number)
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
Number used to approximate the square root of 2
the origin and form uniform angles. A Pell prime is a Pell number that is prime. The first few Pell primes are 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409
Pell_number
Type of prime number
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
Delicate_prime
Natural number
{\displaystyle p} is prime. 32 is the totient summatory function Φ ( n ) {\displaystyle \Phi (n)} over the first 10 integers, and the smallest number n {\displaystyle
32_(number)
Natural number
natural number following 88 and preceding 90. 89 is: the 24th prime number, following 83 and preceding 97. a Chen prime. a Pythagorean prime. the smallest
89_(number)
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