Megaprime

A megaprime is a prime number with at least one million decimal digits.^[1]

Other terms for large primes include "titanic prime", coined by Samuel Yates in the 1980s for a prime with at least 1000 digits^[2] (of which the smallest is 10⁹⁹⁹+7),^[3] and "gigantic prime" for a prime with at least 10,000 digits^[4] (of which the smallest is 10⁹⁹⁹⁹+33603).^[5]

As of 18 April 2024^[update], there are 2,657 known megaprimes^[6] which have more than 1,000,000 digits.^[7] The first to be found was the Mersenne prime 2^6972593−1 with 2,098,960 digits, discovered in 1999 by Nayan Hajratwala, a participant in the distributed computing project GIMPS.^[8]^[9] Nayan was awarded a Cooperative Computing Award from the Electronic Frontier Foundation for this achievement.

Almost all primes are megaprimes, as the number of primes with fewer than one million digits is finite. However, the vast majority of known primes are not megaprimes.

All numbers from 10⁹⁹⁹⁹⁹⁹ through 10⁹⁹⁹⁹⁹⁹ + 593498 are known to be composite, and there is a very high probability that 10⁹⁹⁹⁹⁹⁹ + 593499, a strong probable prime for each of 8 different bases, is the smallest megaprime.^[10] As of 2022^[update], the smallest number known to be a megaprime is 10⁹⁹⁹⁹⁹⁹ + 308267*10²⁹²⁰⁰⁰ + 1.

The last prime that is not a megaprime is almost certainly 10⁹⁹⁹⁹⁹⁹ - 172473.^[7]^{[failed verification]}

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References

^ Chris Caldwell, The Prime Glossary: megaprime at The PrimePages. Retrieved on 2008-01-04.
^ Chris Caldwell, The Prime Glossary: titanic prime at The PrimePages. Retrieved on 2022-06-21.
^ "factordb.com". factordb.com.
^ Chris Caldwell, The Prime Glossary: gigantic prime at The PrimePages. Retrieved on 2022-06-21.
^ "factordb.com". factordb.com.
^ Chris Caldwell, The Largest Known Primes at The PrimePages.
^ ^a ^b Henri Lifchitz & Renaud Lifchitz, Probable Primes Top 10000, primenumbers.net
^ GIMPS press release, GIMPS Finds First Million-Digit Prime. Retrieved on 2008-01-04.
^ Chris Caldwell, The Largest Known Prime by Year: A Brief History at The PrimePages. Retrieved on 2008-09-28.
^ Patrick De Geest, 10^999999 + y, World!Of Numbers

v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers

Large numbers

Examples
in
numerical
order

Expression
methods

Notations	Scientific notation Knuth's up-arrow notation Conway chained arrow notation Steinhaus–Moser notation
Operators	Hyperoperation Tetration Pentation Ackermann function Grzegorczyk hierarchy Fast-growing hierarchy

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