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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
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)
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
Natural number
42 primes 3449 – 90th Sophie Germain prime 3456 – 3-smooth number (27×33) 3457 – Proth prime 3463 – happy number 3467 – safe prime 3469 – super-prime, Cuban
3000_(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
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
9857 – Proth prime 9859 – super-prime 9870 – triangular number 9871 – balanced prime 9880 – tetrahedral number 9887 – safe prime 9901 – unique prime, sum
9000_(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
37. It is a semi-meandric number. 1409 is a super-prime, a Sophie Germain prime, and a Proth prime. 1410 = 2 × 3 × 5 × 47. It is the denominator of the
1000_(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)
Natural number
the third and last trio of three-digit permutable primes in decimal: 199 + 919 + 991 2113 – Proth prime, centered square number 2120 – Fine number 2125
2000_(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
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
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
Natural number
prime number in base 10), following 89 and preceding 101. a Proth prime and a Pierpont prime as it is 3 × 25 + 1. the eleventh member of the Mian–Chowla
97_(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
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
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)
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
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
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
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)
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
Integer named after Reo Fortune
given positive integer n, pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers. For example, to find the seventh
Fortunate_number
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
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
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
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
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
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 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
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
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
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
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
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 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
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
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
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
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
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
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
Primality test for certain numbers
[citation needed] For numbers of the form N = k · 2n + 1 (Proth numbers), either application of Proth's theorem (a Las Vegas algorithm) or one of the deterministic
Lucas–Lehmer–Riesel_test
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
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
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
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 that can be used to count certain kinds of binary trees
Eighth power Perfect power Powerful Prime power Of the form a × 2b ± 1 Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall Other polynomial numbers
Wedderburn–Etherington_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
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
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
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
Type of number introduced by Mike Keith
Eighth power Perfect power Powerful Prime power Of the form a × 2b ± 1 Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall Other polynomial numbers
Keith_number
Sequence of integers
Eighth power Perfect power Powerful Prime power Of the form a × 2b ± 1 Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall Other polynomial numbers
Padovan_sequence
Two raised to an integer power
32 × 15. A prime number that is one less than a power of two is called a Mersenne prime. For example, the prime number 31 is a Mersenne prime because it
Power_of_two
Number whose sums of distinct divisors represent all smaller numbers
determine whether a number is practical from its prime factorization. A positive integer greater than one with prime factorization n = p 1 α 1 . . . p k α k {\displaystyle
Practical_number
Odd composite number which passes the given congruence
odd integer n is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base a, if a and n are coprime, and a ( n − 1
Euler–Jacobi_pseudoprime
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
Type of positive integer
+ 16 each share a prime factor with one of 2184 = 23 · 3 · 7 · 13 and 2200 = 23 · 52 · 11. These 15 numbers and their shared prime factor(s) are: The
Erdős–Woods_number
Integer having only small prime factors
is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number in which every prime factor is at most 7
Smooth_number
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
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
Mathematical concept in prime numbers
in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power. In particular, a number that has two distinct
Idoneal_number
Primality test for Fermat numbers
which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named after a French mathematician, Théophile
Pépin's_test
Class of natural numbers with many divisors
possible number of divisors for the number of (not necessarily distinct) prime factors it has. An effective construction of the set of all superior highly
Superior highly composite number
Superior_highly_composite_number
Number that remains the same when its digits are reversed
possess a certain property and are palindromic. For instance: The palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, ... (sequence A002385 in the OEIS). The
Palindromic_number
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME
PROTH PRIME