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Prime number of the form k*(2^n)+1
2^{n}>k} . A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth. The first few Proth primes are 3, 5
Proth_prime
Primality test for numbers of a certain form
Proth's theorem is a theorem which forms the basis of a primality test for Proth numbers known as Proth's test. Proth numbers, sometimes called Proth
Proth's_theorem
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
Number divisible only by 1 and itself
specific number forms include Pépin's test for Fermat numbers (1877), Proth's theorem (c. 1878), the Lucas–Lehmer primality test (originated 1856), and
Prime_number
French mathematician
famous of these, Proth's theorem, can be used to test whether a Proth number (a number of the form k × 2n + 1 with k odd and k < 2n) is prime. The numbers
François_Proth
Natural number
super-prime, a Fibonacci prime, a Markov prime, and an emirp. 1601 is a Sophie Germain prime and a Proth prime. 1607, 1609, and 1613 form a prime triple
1000_(number)
BOINC based volunteer computing project researching prime numbers
year, PrimeGrid migrated its systems from PerlBOINC to standard BOINC software. Since September 2008, PrimeGrid is also running a Proth prime sieving
PrimeGrid
Natural number
{41, 83, 167}. It is an Eisenstein prime, with no imaginary part and real part of the form 3n − 1. It is a Proth prime because 41 = 5 × 23 + 1. It is the
41_(number)
Prime number of the form 2^n – 1
Mersenne Prime Search (GIMPS) Largest known prime number Wieferich prime Wagstaff prime Cullen prime Woodall prime Proth prime Solinas prime Gillies'
Mersenne_prime
Natural number
prime number, a happy prime, and an emirp with 79. It is the smallest odd prime that is not a cluster prime. Because 97 = 3 × 25 + 1, 97 is a Proth prime
97_(number)
Natural number
is a Proth prime and a centered square number. 2116 = 462 = 22 × 232. 2125 = 53 × 17. It is a nonagonal number. 2129 is a Sophie Germain prime. 2141
2000_(number)
Natural number
in ternary (110220113), tribonacci number 3137 – Proth prime, both a left- and right-truncatable prime 3149 – highly cototient number 3155 – member of
3000_(number)
Positive integer of the form (2^(2^n))+1
partially depends on Fermat primes. Double exponential function Lucas' theorem Mersenne prime Pierpont prime Primality test Proth's theorem Pseudoprime Sierpiński
Fermat_number
Natural number
9857 – Proth prime 9859 – super-prime 9870 – triangular number 9871 – balanced prime 9880 – tetrahedral number 9887 – safe prime 9901 – unique prime, sum
9000_(number)
Natural number
square mile. 641 is a prime number, a Sophie Germain prime, a Chen prime, an Eisenstein prime with no imaginary part and a Proth prime. It is a factor of
600_(number)
largest known primes are Mersenne primes. The last 18 record primes were Mersenne primes. The binary representation of any Mersenne prime is composed of
Largest_known_prime_number
Natural number
a lucky prime. Since 241 = 15 × 24 + 1, it is a Proth prime. 241 is a repdigit in base 15 (111). 241 is the only known Lucas–Wieferich prime to (U, V)
241_(number)
Prime number of the form that allows fast modular reduction
which has also been called pseudo-Mersenne. Proth prime: several examples on this page are also Proth primes Solinas, Jerome A. (1999). Generalized Mersenne
Solinas_prime
Prime number of the form 2^u × 3^v + 1
of Mersenne primes in that range. When 2 u > 3 v {\displaystyle 2^{u}>3^{v}} , 2 u ⋅ 3 v + 1 {\displaystyle 2^{u}\cdot 3^{v}+1} is a Proth number and thus
Pierpont_prime
Natural number
There are 576 parts in all compositions of 8. 577 is a prime number, a Proth prime, and a Chen prime. It is palindromic in bases 18 (1E118) and 24 (10124)
500_(number)
Natural number
353 is the 71st prime number, a palindromic prime, an irregular prime, a super-prime, a Chen prime, a Proth prime, and an Eisenstein prime. In connection
353_(number)
Natural number
prime number, a Proth prime, a palindromic prime, an Eisenstein prime with no imaginary part, and the sum of nine consecutive primes (83 + 89 + 97 + 101
900_(number)
Natural number
eight consecutive primes (79 + 83 + 89 + 97 + 101 + 103 + 107 + 109). 769 is a prime number, a Chen prime, a lucky prime, and a Proth prime. 770 = 2 × 5 ×
700_(number)
Natural number
of primes ≤ 2 17 {\displaystyle \leq 2^{17}} 12285 = amicable number with 14595 12287 = Thabit number 12288 = 3-smooth number (212×3). 12289 = Proth prime
10,000
Natural number
number. 449 is a prime number, a Chen prime, a Proth prime, an Eisenstein prime with no imaginary part, and the sum of five consecutive primes (79 + 83 + 89
400_(number)
reciprocals of the Proth primes, of which there may be finitely many or infinitely many, is known to be finite, approximately 0.747392479. The prime quadruplets
List_of_sums_of_reciprocals
Natural number
having clique number 4. The number 301*2^184+1 is the first, and smallest Proth prime with k = 301. 301 is a centered tridecagonal number. The matula tree
301_(number)
Natural number
a prime number, a Sophie Germain prime, an emirp, an isolated prime, a Chen prime, a highly cototient number, a centered square number, and a Proth prime
113_(number)
Type of prime number conjectured to exist
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
Wall–Sun–Sun_prime
Probabilistic primality test
Mersenne primes, Mersenne cofactors, and Proth primes; Li's modification generalizes it to any n. The GIMPS in particular tests Mersenne primes and Mersenne
Fermat_primality_test
Numbers with a certain property involving recursive summation
{\displaystyle b} -happy prime will not necessarily create another happy prime. For instance, while 19 is a 10-happy prime, 91 = 13 × 7 is not prime (but is still
Happy_number
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
Decomposition of a number into a product
until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem
Integer_factorization
Odd number with specific properties
be the five smallest). Mathematics portal Cullen number Proth number Woodall number "The Prime Glossary: Sierpinski number". t5k.org. Retrieved 2026-06-29
Sierpiński_number
Integer filtered out using a sieve similar to that of Eratosthenes
This sieve is similar to the sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set
Lucky_number
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
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
Conjecture in number theory
years before Gilbreath's discovery, François Proth had published the same observations. Consider the prime numbers 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 ,
Gilbreath's_conjecture
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
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
Open source middleware system for volunteer and grid computing
Cyberscience Centre distributed.net Folding@home Great Internet Mersenne Prime Search grid.org Gridcoin BOSSA "BOINC License". GitHub. Archived from the
Berkeley Open Infrastructure for Network Computing
Berkeley_Open_Infrastructure_for_Network_Computing
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
Numbers with many divisors
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 a smaller prime, and
Highly_composite_number
scientist) (2008-06-06). "PrimeGrid's Birthday Challenge". Retrieved 2012-01-29. "PrimeGrid". 2012. Retrieved 2012-01-13. "BOINCstats — PrimeGrid". boincstats
List of volunteer computing projects
List_of_volunteer_computing_projects
Power of a prime number
a prime power is a positive integer that is a positive integer power of a single prime number. For example: 7 = 71, 9 = 32 and 64 = 26 are prime powers
Prime_power
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
Odd number with specific properties
number List of primes of the form: k*2^n-1, k<300 List of primes of the form: k*2^n-1, k<300, Project Riesel Prime Search Riesel and Proth Prime Database
Riesel_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
Number that is the result of operation on its own digits
recreational mathematics enthusiast. A Friedman prime is a Friedman number that is also prime. The decimal Friedman primes are: 127, 347, 2503, 12101, 12107, 12109
Friedman_number
Probabilistic test for the primality of an integer
Fibonacci pseudoprimes are composite integers that pass certain tests which all primes and very few composite numbers pass: in this case, criteria relative to
Lucas_pseudoprime
Product of two prime numbers
product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there
Semiprime
Natural number with a decimal representation made of repeated instances of the same digit
repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary). Any
Repdigit
Concatenation of the first n prime numbers
Smarandache–Wellin number is a prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein as a probable prime in 1998 and then proven prime in 2022. In March
Smarandache–Wellin_number
Probable prime that is composite
pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified
Pseudoprime
Centered figurate number
superstar prime is a star prime whose prime index is also a star number. The first two such numbers are 661 and 1750255921. A reverse superstar prime is a
Star_number
Algorithm for determining whether a number is prime
known that PRIMES is not in AC0. Certain number-theoretic methods exist for testing whether a number is prime, such as the Lucas test and Proth's test. These
Primality_test
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
Numbers k where x - phi(x) = k has many solutions
least one prime factor in common with x {\displaystyle x} . For example, the cototient of 6 is 4 since these four positive integers have a prime factor in
Highly_cototient_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
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 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
Mathematical concept
Proth numbers. In 1976 Christopher Hooley showed that the natural density of positive integers n ≤ x {\displaystyle n\leq x} for which Cn is a prime is
Cullen_number
Number equal to the sum of its proper divisors
to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2 p − 1 {\displaystyle 2^{p}-1} with a prime p are prime; for
Perfect_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
Proof that a number is prime
{\displaystyle P} is prime. Gerbicz-based certificate seek to prove the correctness of a modular exponentiation process as used in the Proth and Fermat probabilistic
Primality_certificate
Integer having a non-trivial divisor
positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the natural numbers that are not prime and not a unit. For example
Composite_number
Integer named after Reo Fortune
p_{n}\#+m} is a prime number, where the primorial p n # {\displaystyle p_{n}\#} is the product of the first n {\displaystyle n} prime numbers. They are
Fortunate_number
Numbers in a type of Lucas sequence
(sequence A001045 in the OEIS) A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are: 3, 5, 11, 43, 683, 2731, 43691,
Jacobsthal_number
Class of numbers in number theory
numbers base 2 are exactly the Mersenne numbers. A Williams prime is a Williams number that is prime. They were considered by Hugh C. Williams. It is conjectured
Williams_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
Number of the digit form ABABAB... and A is not equal to B
Undulating numbers with odd number of digits are palindromic. They can be prime, for example 151. The undulating number ABAB...AB with n repetitions of
Undulating_number
Numeral ambigram
upside down (e.g., 69, 96, 1001). A strobogrammatic prime is a strobogrammatic number that is also a prime number, i.e., a number that is only divisible by
Strobogrammatic_number
Number with few prime factors
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 the
Almost_prime
Infinite integer series where the next number is the sum of the two preceding it
L5466311, with 1,142,392 decimal digits. If Ln is prime then n is 0, prime, or a power of 2. L2m is prime for m = 1, 2, 3, and 4 and no other known values
Lucas_number
Composite number which passes Miller–Rabin primality test
is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making
Strong_pseudoprime
Odd composite number which passes the given congruence
that all prime numbers p satisfy the above equation which can be deduced from Fermat's little theorem. Fermat's theorem asserts that if p is prime, and coprime
Euler_pseudoprime
Type of natural number in recreational number theory
the number of prime numbers which can be obtained by permuting some or all of its digits (in base 10) is larger than the number of primes obtainable in
Primeval_number
Sequence of integers
power Perfect power Powerful Prime power Of the form a × 2b ± 1 Cunningham Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall Other polynomial
Padovan_sequence
Count of the possible partitions of a set
whether infinitely many Bell numbers are also prime numbers. These are called Bell primes. The first few Bell primes are: 2, 5, 877, 27644437,
Bell_number
Type of natural number
... (sequence A003052 in the OEIS) A self prime is a self number that is prime. The first few self primes in base 10 are 3, 5, 7, 31, 53, 97, 211, 233
Self_number
Type of number introduced by Mike Keith
power Perfect power Powerful Prime power Of the form a × 2b ± 1 Cunningham Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall Other polynomial
Keith_number
Odd composite number which passes the given congruence
In number theory, an odd integer n is called an Euler–Jacobi probable prime to base a, if a and n are coprime, and a ( n − 1 ) / 2 ≡ ( a n ) ( mod n )
Euler–Jacobi_pseudoprime
Positive integer that is the product of three distinct prime numbers
integer that is the product of three distinct prime numbers. For example, since 2, 3, and 73 are all prime, 438 is a sphenic number because 2 × 3 ×
Sphenic_number
Positive integer of the form 4n + 1
A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since 21 = 3 ⋅ 7. However, 21 is a Hilbert prime since neither
Hilbert_number
Mathematical concept
k < n, the polynomial k2 − k + n produces a prime number. When k is equal to n, the value cannot be prime since n2 − n + n = n2 is divisible by n. Since
Lucky_numbers_of_Euler
Product of two distinct primes ≡ 3 (mod 4)
Blum integer if n = p × q is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. That is, p and q must be of the form 4t +
Blum_integer
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
Composite number that passes Fermat's probable primality test
theorem. Fermat's little theorem states that if p {\displaystyle p} is prime and a {\displaystyle a} is coprime to p {\displaystyle p} , then a p − 1
Fermat_pseudoprime
Number-theoretic algorithm
(a strengthening of an 1878 theorem of Proth): Let N − 1 = m p {\displaystyle N-1=mp} where p is an odd prime such that 2 p + 1 > N {\displaystyle 2p+1>{\sqrt
Pocklington n − 1 primality test
Pocklington_n_−_1_primality_test
Number of unique ways to draw non-intersecting chords in a circle
{3}{n}}\right)^{3/2}3^{n},~n\to \infty } . A Motzkin prime is a Motzkin number that is prime. Four such primes are known: 2, 127, 15511, 953467954114363 (sequence
Motzkin_number
Product of the first "n" prime numbers
the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial"
Primorial
Set of numbers used in the smoothsort algorithm
and also analyzed them in some detail. A Leonardo prime is a Leonardo number that is also prime. The term "Leonardo number" was coined by Dijkstra,
Leonardo_number
Number of ways to pair up n objects
of two in the prime factorization) of T(4k) and of T(4k + 1) is k; for T(4k + 2) it is k + 1, and for T(4k + 3) it is k + 2. For any prime number p, one
Telephone number (mathematics)
Telephone_number_(mathematics)
Numbers obtained by adding the two previous ones
{x^{k}}{k!}}+\sum _{k=0}^{\infty }F_{k}{\frac {x^{k}}{k!}}\\F^{\prime \prime }(x)={}&F^{\prime }(x)+F(x)\end{aligned}}} The characteristic polynomial of this
Fibonacci_sequence
Number that can be used to count certain kinds of binary trees
power Perfect power Powerful Prime power Of the form a × 2b ± 1 Cunningham Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall Other polynomial
Wedderburn–Etherington_number
Type of Poulet number
get super-Poulet numbers with 3 distinct prime divisors. If you find three Poulet numbers with three common prime factors, you get a super-Poulet number
Super-Poulet_number
Function whose domain is the positive integers
number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes
Arithmetic_function
Natural number
first 100 cubed numbers 27,644,437 = Bell number 31,172,165 = Smallest Proth exponent for n = 10223 (see Seventeen or Bust) 31,536,000 = Standard number
10,000,000
Iterative algorithm on numbers
power Perfect power Powerful Prime power Of the form a × 2b ± 1 Cunningham Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall Other polynomial
Kaprekar's_routine
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PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
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