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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
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
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
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
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
Topics referred to by the same term
known prime number, the record-holder currently or historically Mersenne prime This disambiguation page lists articles associated with the title Largest prime
Largest_prime
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
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 nth Fibonacci number. Largest known prime number Caldwell, Chris K. "THE LARGEST KNOWN PRIMES (The 5,000 largest known primes)". Retrieved 23 November
List of largest known primes and probable primes
List_of_largest_known_primes_and_probable_primes
Natural number
the first prime gap sequence of 2, 4, 6, 8, 10, 12, 14 129,106 = Keith number 130,321 = 194 131,071 = Mersenne prime 131,072 = 217, largest determinant
100,000
Prime number with at least one million digits
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
Megaprime
Natural number
006,003 = largest known generalized Wieferich prime in base 3, and only one apart from 11 1,006,301 = First number of the first pair of prime quadruplets
1,000,000
Natural number
is the natural number following 60 and preceding 62. 61 is the 18th prime number, and a twin prime with 59. As a centered square number, it is the sum
61_(number)
Musical instrument tuning with a limit of seven
is a musical tuning where the largest prime number factor of the interval ratios between pitches is seven. The only primes available in septimal tuning
7-limit_tuning
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
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
Positive integer of the form (2^(2^n))+1
there are no known prime factors for n = 20 and n = 24. The largest Fermat number known to be composite is F18233954, and its prime factor 7 × 218233956
Fermat_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
Natural number
the largest prime factor of the five-digit repunit 11111, and the largest prime number for which the decimal period of its multiplicative inverse is 5:
271_(number)
Head of government
the leader of the political party that holds the largest number of seats in the Commons. The prime minister is ex officio also First Lord of the Treasury
Prime Minister of the United Kingdom
Prime_Minister_of_the_United_Kingdom
Difference between two successive prime numbers
to the average prime gap sizes in the vicinity of p n {\displaystyle p_{n}} . The largest known prime gap with identified probable prime gap ends has length
Prime_gap
Russian indie music group
Prostoe Chislo (Russian: Самое Большое Простое Число, lit. 'The Largest Prime Number') is a Russian indie group formed in 2006 by Kirill Ivanov and the
Samoe_Bolshoe_Prostoe_Chislo
Natural number
+ (−3)3. The smallest number that can be written as the sum of two positive cubes in two ways is 1729. The largest prime number that can be represented
189_(number)
Natural number
zeta function, ζ . {\displaystyle \zeta .} 29 is the largest prime factor of the smallest number with an abundancy index of 3, 1018976683725 = 33 × 52
29_(number)
fifth and largest known unitary perfect number. Mathematics: 357,686,312,646,216,567,629,137 (≈3.6×1023) is the largest left-truncatable prime. Mathematics:
Orders_of_magnitude_(numbers)
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
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
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)
Computer checksum algorithm
is initialized to 1, B to 0. The sums are done modulo 65521 (the largest prime number smaller than 216). The bytes are stored in network order (big endian)
Adler-32
Natural number
record-holding prime, 6,700,417 – also discovered by Euler – forty years earlier. The number 2,147,483,647 remained the largest known prime until 1867. In
2,147,483,647
Natural number
shift register; also number of binary irreducible polynomials whose degree divides 32 136,279,841 = The largest known Mersenne prime exponent, as of October
100,000,000
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
Natural number
also a Wagstaff prime, a Gaussian prime, and a Heegner number. 43 is the fourth term of Sylvester's sequence. 43 is the largest prime which divides the
43_(number)
Number equal to the sum of its proper divisors
2019). "Upper bounds on the second largest prime factor of an odd perfect number". International Journal of Number Theory. 15 (6): 1183–1189. arXiv:1810
Perfect_number
Set of prime numbers linked by a linear relationship
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression
Primes in arithmetic progression
Primes_in_arithmetic_progression
Topics referred to by the same term
claimed as the largest number ever used in a serious mathematical proof Largest known prime number, for the largest known primes Dirac large numbers hypothesis
Large_number_(disambiguation)
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
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
179 = largest number such that every pair of consecutive digits is a distinct prime. 627,013,566,048 = 32nd Pell number. 649,164,795,179 = number of centered
100,000,000,000
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
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)
Natural number
largest of the four all-Harshad numbers. 6 is the 2nd superior highly composite number, the 2nd colossally abundant number, the 3rd triangular number
6
Natural number
also the largest known Mersenne prime exponent for a Mersenne number, 2 127 − 1 {\displaystyle 2^{127}-1} , which is also a Mersenne prime. It was discovered
127_(number)
Natural number
9,876,543,210 : largest pandigital number. 9,999,800,001 = 999992, the largest ten-digit square. 9,999,999,967 : greatest prime number with 10 digits "Definition
1,000,000,000
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)
Natural number
radicals. Four is the only square number N = n × n {\displaystyle N=n\times n} where N − 1 {\displaystyle N-1} is a prime number. The four-color theorem states
4
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)
Natural number
3087 – sum of first 40 primes 3109 – super-prime 3119 – safe prime 3121 – centered square number, emirp, largest minimal prime in quinary. 3125 – a solution
3000_(number)
Used to count, measure, and label
October 2025. Caldwell, Chris. "The Largest Known prime by Year: A Brief History". The PrimePages: prime number research & records. Archived from the
Number
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
BOINC based volunteer computing project researching prime numbers
Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available
PrimeGrid
Natural number
(seventeen) is the natural number following 16 and preceding 18. It is a prime number. 17 is a Leyland number and Leyland prime, using 2 and 3 (23 + 32)
17_(number)
Volunteer project using software to search for Mersenne prime numbers
has found 18 Mersenne primes, 16 of which were the largest known prime number when they were discovered. The largest known prime is 2136,279,841 − 1 (or
Great Internet Mersenne Prime Search
Great_Internet_Mersenne_Prime_Search
Residence and office of the UK prime minister
residence and office of the prime minister of the United Kingdom, as the first lord of the treasury. Colloquially known as Number 10, the building is located
10_Downing_Street
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
Primality test for numbers of a certain form
"PrimePage Primes: The Top Twenty". t5k.org. Retrieved 2026-06-29. "World Record Colbert Number discovered!". "PrimePage Primes: Largest Known Primes"
Proth's_theorem
Head of government
from the Knesset to take office. In practice, the prime minister is usually the leader of the largest party in the governing coalition. Since it is all
Prime_Minister_of_Israel
Number of the form (n * 2^n) - 1
Woodall number W2m may be prime only if 2m + m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W2
Woodall_number
French mathematician (1842–1891)
is prime; it remained the largest known Mersenne prime for three-quarters of a century, and remains the largest prime number proved by hand. Later Derrick
Édouard_Lucas
Prime number of the form 2^u × 3^v + 1
In number theory, a Pierpont prime is a prime number of the form 2 u ⋅ 3 v + 1 {\displaystyle 2^{u}\cdot 3^{v}+1\,} for some nonnegative integers u and
Pierpont_prime
Prime number of the form k*(2^n)+1
238411} As of June 2023, the largest Proth primes discovered are: A Proth number of the second kind is a natural number N of the form N = k × 2 n − 1
Proth_prime
Proof method
{\displaystyle n/p_{k}} , where p k {\displaystyle p_{k}} is the largest prime number dividing n {\displaystyle n} . Since n {\displaystyle n} is incompressible
Incompressibility_method
Even integers as sums of two primes
unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture
Goldbach's_conjecture
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)
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
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
Natural number
quadratic fields with class number of 7, negated (see also Heegner numbers). 71 is the largest number which occurs as a prime factor of an order of a sporadic
71_(number)
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
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
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
largest such (Heegner) number of class one. The twenty-first prime number 73 is the largest member of Bhargava's definite quadratic 17–integer matrix Φ
21_(number)
Natural number
hundred [and] three) is the natural number following 102 and preceding 104. 103 is a prime number, and the largest prime factor of 6 ! + 1 = 721 = 7 ⋅ 103
103_(number)
political parties, the prime minister is usually the leader of the largest party represented in the house. Since 1935, every prime minister has been a member
List of prime ministers of New Zealand
List_of_prime_ministers_of_New_Zealand
Set of 3 prime numbers whose largest and smallest differ by 6
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
Prime_triplet
Head of government of the Republic of India
The prime minister of India (ISO: Bhāratīya Gaṇarājyasya Pradhānamantrī) is the head of government of the Republic of India. Executive authority is vested
Prime_Minister_of_India
Head of government
the prime minister typically sits as a member of Parliament (MP) and leads the largest party or a coalition of parties. As first minister, the prime minister
Prime_Minister_of_Canada
Result on density of prime numbers
. In number theory, Bertrand's postulate is the theorem that for any integer n > 3 {\displaystyle n>3} , there exists at least one prime number p {\displaystyle
Bertrand's_postulate
Indian film producer and visual effects executive (born 1976)
director of Indian visual and special effects company Prime Focus Limited, the world's largest independent integrated media services company, and the
Namit_Malhotra
Natural number
eleven) is the natural number following 3510 and preceding 3512. 3511 is a prime number, and is also an emirp: a different prime when its digits are reversed
3511
Number of the form x^y + y^x
Leyland primes, for example x2 + 2x is prime for x = 3, 9, 15, 21, 33, 2007, 2127, 3759, ... (OEIS: A064539). By November 2012, the largest Leyland number that
Leyland_number
In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than n {\displaystyle {\sqrt {n}}} . A k-smooth
Unusual_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
Natural number
store in memory. Since the prime factors of 232 − 1 are exactly the five known Fermat primes, this number is the largest known odd value n for which
4,294,967,295
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
There is a prime between any two square numbers
O({\sqrt {p}}\log p)} on the size of the largest prime gaps. By the prime number theorem, the expected number of primes between n 2 {\displaystyle n^{2}} and
Legendre's_conjecture
Natural number
The largest five-digit number that is a left-and-right-truncatable prime 99,856 = 3162, the largest 5-digit square 99,991 = largest five-digit prime number
90,000
for the 2014 Winter Olympics. Two buildings: Mandalay Bay and W. "Largest hotel (number of rooms)". Guinness World Records. Retrieved 2019-09-22. "Compare
List_of_largest_hotels
Natural number
of integers has a unique factorization, or class number of 1. 9 is the largest single-digit number in the decimal system. The nine dots puzzle is a mathematical
9
Type of prime number
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
Permutable_prime
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
Minimal prime (recreational mathematics)
Minimal_prime_(recreational_mathematics)
Natural number
Fermat primes is equal to the number of sides of the largest known regular constructible polygon with a straightedge and compass that has an odd number of
32_(number)
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
Country in northwestern Europe
the largest total number of users in that year. The British video game industry is the largest in Europe, and since 2022 the UK has the largest video
United_Kingdom
This article lists the largest shopping malls in the Philippines by gross floor area. SM Prime Holdings is the largest shopping retail operator in the
List of largest shopping malls in the Philippines
List_of_largest_shopping_malls_in_the_Philippines
Natural number
26 cells 9,999,999,999,971 : largest 13-digit prime number 9,999,999,999,999 : largest 13-digit number, repdigit Long and short scales Names of large
1,000,000,000,000
to the number itself. 7, the first and only prime number preceding a cube. 8, the largest cube in the Fibonacci sequence. 9, the first odd number that is
List_of_numbers
Solution to x*x + y*y + z*z = 3xyz
Schiffler in 2023. None of the prime divisors of a Markov number is congruent to 3 modulo 4, which implies that an odd Markov number is 1 more than a multiple
Markov_number
Line of longitude, at which longitude is defined to be 0°
prime meridian is a meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. On a spheroid, a prime meridian
Prime_meridian
Head of government of Morocco
appointed by the king of Morocco from the political party that wins the largest number of seats in the House of Representatives, usually the general secretary
Prime_Minister_of_Morocco
Natural number
number following 266 and preceding 268. 267 is an odd composite number with two prime factors. 267 is the number of planar partitions of the number 12
267_(number)
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