# Category: Prime numbers

Generation of primes
In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently. These are used in various applications, for example hashing, public-key cryptography, an
Prime constant
The prime constant is the real number whose th binary digit is 1 if is prime and 0 if is composite or 1. In other words, is the number whose binary expansion corresponds to the indicator function of t
Copeland–Erdős constant
The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value, using the modern definition of prime, is approximately 0.23571113171
Pillai sequence
The Pillai sequence is the sequence of integers that have a record number of terms in their greedy representations as sums of prime numbers (and one).It is named after Subbayya Sivasankaranarayana Pil
Wike's law of low odd primes
Wike's law of low odd primes is a methodological principle to help design sound experiments in psychology. It is: "If the number of experimental treatments is a low odd prime number, then the experime
Ulam spiral
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in S
Riesel number
In mathematics, a Riesel number is an odd natural number k for which is composite for all natural numbers n (sequence in the OEIS). In other words, when k is a Riesel number, all members of the follow
Prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by π(x) (unrelated to the number π).
Freshman's dream
The freshman's dream is a name sometimes given to the erroneous equation , where is a real number (usually a positive integer greater than 1) and are nonzero real numbers. Beginning students commonly
Megaprime
A megaprime is a prime number with at least one million decimal digits. Other terms for large primes include titanic prime, coined by Samuel Yates in the 1980s for a prime with at least 1000 digits, a
Fermi–Dirac prime
In number theory, a Fermi–Dirac prime is a prime power whose exponent is a power of two. These numbers are named from an analogy to Fermi–Dirac statistics in physics based on the fact that each intege
SuperPrime
SuperPrime is a computer program used for calculating the primality of a large set of positive natural numbers. Because of its multi-threaded nature and dynamic load scheduling, it scales excellently
43,112,609
43,112,609 (forty-three million, one hundred twelve thousand, six hundred nine) is the natural number following 43,112,608 and preceding 43,112,610.
PrimePages
The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller
Interprime
In mathematics, an interprime is the average of two consecutive odd primes. For example, 9 is an interprime because it is the average of 7 and 11. The first interprimes are: 4, 6, 9, 12, 15, 18, 21, 2
Gaussian moat
In number theory, the Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequ
Furstenberg's proof of the infinitude of primes
In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined clos
Euclid–Mullin sequence
The Euclid–Mullin sequence is an infinite sequence of distinct prime numbers, in which each element is the least prime factor of one plus the product of all earlier elements. They are named after the
Prime k-tuple
In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a k-tuple (a, b, …), the positions where the k-tuple matc
Prime signature
In mathematics, the prime signature of a number is the multiset of (nonzero) exponents of its prime factorization. The prime signature of a number having prime factorization is the multiset . For exam
Selberg's identity
In number theory, Selberg's identity is an approximate identity involving logarithms of primes found by Selberg. Selberg and Erdős both used this identity to give elementary proofs of the prime number
Bi-twin chain
In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers in which every number is prime. The numbers form a Cunningham chain of the first kind of length , while forms a Cunni
The Music of the Primes
The Music of the Primes (British subtitle: Why an Unsolved Problem in Mathematics Matters; American subtitle: Searching to Solve the Greatest Mystery in Mathematics) is a 2003 book by Marcus du Sautoy
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
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively
Fortunate number
A Fortunate number, named after Reo Fortune, is the smallest integer m > 1 such that, for a given positive integer n, pn# + m is a prime number, where the primorial pn# is the product of the first n p
Ruth–Aaron pair
In mathematics, a Ruth–Aaron pair consists of two consecutive integers (e.g., 714 and 715) for which the sums of the prime factors of each integer are equal: 714 = 2 × 3 × 7 × 17,715 = 5 × 11 × 13, an
List of largest known primes and probable primes
The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the University of Tennessee's PrimePages and Henri & Renaud Lifchitz' PRP Records. Numbers with
Reciprocals of primes
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737. Like all rational numbers, the reciprocal
Composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself.
Formula for primes
In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are k
Largest known prime number
The largest known prime number (as of November 2022) is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the
Table of prime factors
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 in bold below. The number 1 is called a un
Prime gap
A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n + 1)-th and then-th prime numbers, i.e. We have g1 = 1, g2
Primes in arithmetic progression
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7,
Provable prime
In number theory, a provable prime is an integer that has been calculated to be prime using a primality-proving algorithm. Boot-strapping techniques using Pocklington primality test are the most commo
Wieferich pair
In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2) Wieferich pairs are named after German mathematician Arthur Wieferich.Wiefe
Chebyshev's bias
In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed b
Mills' constant
In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function is a prime number for all natural numbers n. This co
Prime omega function
In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby (little omega) counts each distinct prime factor, whereas the related function (big omega)
Bertrand's postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer , there always exists at least one prime number with A less restrictive formulation is: for every , there is always at
Industrial-grade primes (the term is apparently due to Henri Cohen) are integers for which primality has not been certified (i.e. rigorously proven), but they have undergone probable prime tests such
Primecoin
Primecoin (Abbreviation: XPM; sign: Ψ) is a cryptocurrency that implements a proof-of-work system that searches for chains of prime numbers.
Almost prime
In number theory, a natural number is called k-almost prime if it has k prime factors. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in
Cunningham chain
In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes.
Sierpiński number
In number theory, a Sierpiński number is an odd natural number k such that is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which
Prime power
In mathematics, a prime power is a positive integer which is a power of a single prime number.For example: 7 = 71, 9 = 32 and 64 = 26 are prime powers, while6 = 2 × 3, 12 = 22 × 3 and 36 = 62 = 22 × 3
Lunar arithmetic
Lunar arithmetic, formerly called dismal arithmetic, is a version of arithmetic in which the addition and multiplication operations on digits are defined as the max and min operations. Thus, in lunar
Prime number
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 is not prime is called a composite number. For
Semiprime
In mathematics, 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, so the semiprimes include the squares of prime num
Proof of Bertrand's postulate
In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Chebyshev, and a short but advanced proof was given by Ramanujan. The
Closing the Gap: The Quest to Understand Prime Numbers
Closing the Gap: The Quest to Understand Prime Numbers is a book on prime numbers and prime gaps by Vicky Neale, published in 2017 by the Oxford University Press (ISBN 9780198788287). The Basic Librar
Primeval number
In recreational number theory, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all of its digits (in base 10) is larger than th
Goldbach's comet
Goldbach's comet is the name given to a plot of the function , the so-called Goldbach function (sequence in the OEIS). The function, studied in relation to Goldbach's conjecture, is defined for all ev
Backhouse's constant
Backhouse's constant is a mathematical constant named after Nigel Backhouse. Its value is approximately 1.456 074 948. It is defined by using the power series such that the coefficients of successive
Smarandache–Wellin number
In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after and .
Sphenic number
In number theory, a sphenic number (from Ancient Greek: σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, ther
Division lattice
The division lattice is an infinite complete bounded distributive lattice whose elements are the natural numbers ordered by divisibility. Its least element is 1, which divides all natural numbers, whi
Lucky numbers of Euler
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k is equal to n, the value cannot be prime since n2
Belphegor's prime
Belphegor's prime is the palindromic prime number 1000000000000066600000000000001 (1030 + 666 × 1014 + 1), a number which reads the same both backwards and forwards and is only divisible by itself and