"the first nine prime numbers, as well as the sum of some pairs of prime numbers e.g., 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, and 47 + 53. 100 is the sum of the cubes of the first four integers (100 = 1 + 2 + 3 + 4). This is related by Nicomachus's theorem to the fact that 100 also equals the square of the sum of the first four integers: . 2 + 6 = 100, thus 100 is a Leyland number. 100 is an 18-gonal number. It is divisible"
"the number of primes with less than a million digits is finite. However, the vast majority of known primes are not megaprimes. Entries labelled ""Prime"" have been proved prime; those labelled ""PRP"" have not. See the article Probable prime. There are currently 412 proven mega primes. All numbers from 10 through 10 + 593498 are known to be composite, and there is a very high probability 10 + 593499, a strong PRP, is actually the smallest megaprime. formula_1 = gcd(formula_2,the aurifeuillean ""L"" part of formula_3), formula_4 = gcd(formula_2,the aurifeuillean ""M"" part of formula_3). Megaprime A megaprime is a prime number"
"is the only triplet that can ever be prime. 111 is not prime in base ten, but is prime in base two, where 111 = 7. It is also prime in these other bases up to 128: 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119 In base 18, the number 111 is 7 (= 343) which is the only base where 111 is a perfect power. The smallest magic square using only"
"277 (number) 277 (two hundred [and] seventy-seven) is the natural number following 276 and preceding 278. 277 is the 59th prime number, and is a regular prime. It is the smallest prime ""p"" such that the sum of the inverses of the primes up to ""p"" is greater than two. Since 59 is itself prime, 277 is a super-prime. 59 is also a super-prime (it is the 17th prime), as is 17 (the 7th prime). However, 7 is the fourth prime number, and 4 is not prime. Thus, 277 is a super-super-super-prime but not a super-super-super-super-prime. It is the largest"
"by 25, the number of primes below it. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. It can be expressed as a sum of some of its divisors, making it a semiperfect number. 100 is a Harshad number in base 10, and also in base 4, and in that base it is a self-descriptive number. There are exactly 100 prime numbers whose digits are in strictly ascending order (e.g. 239, 2357 etc.). 100 is the smallest number whose common logarithm is a prime number (i.e. 10"
"2 − 1. It has 13,395 digits and was found in 1979 by Harry L. Nelson and David Slowinski. The smallest gigantic prime is 10 + 33603. It was proved prime in 2003 by Jens Franke, Thorsten Kleinjung and Tobias Wirth with their own distributed ECPP program. It was the largest ECPP proof at the time. Gigantic prime A gigantic prime is a prime number with at least 10,000 decimal digits. The term appeared in ""Journal of Recreational Mathematics"" in the article ""Collecting gigantic and titanic primes"" (1992) by Samuel Yates. Chris Caldwell, who continued Yates' collection in The Prime"
"Largest known prime number The largest known prime number () is 2 − 1, a number with 23,249,425 digits. It was found by the Great Internet Mersenne Prime Search (GIMPS) in 2017. Euclid proved that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers. Many of the largest known primes are Mersenne primes. , the seven largest known primes are Mersenne primes. The last sixteen record primes were Mersenne primes. The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests"
"Prime number A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 6 is composite because it is the product of two numbers () that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either"
"this way. One hundred and eighteen is also: 118 (number) 118 (one hundred [and] eighteen) is the natural number following 117 and preceding 119. There is no answer to the equation φ(x) = 118, making 118 a nontotient. 118 is the smallest ""n"" such that the range ""n"", ""n"" + 1, ... 4""n""/3 contains at least one prime from each of these forms: 4""x"" + 1, 4""x"" - 1, 6""x"" + 1 and 6""x"" - 1. Four expressions for 118 as the sum of three positive integers have the same product: 118 is the smallest number that can be expressed"
"118 (number) 118 (one hundred [and] eighteen) is the natural number following 117 and preceding 119. There is no answer to the equation φ(x) = 118, making 118 a nontotient. 118 is the smallest ""n"" such that the range ""n"", ""n"" + 1, ... 4""n""/3 contains at least one prime from each of these forms: 4""x"" + 1, 4""x"" - 1, 6""x"" + 1 and 6""x"" - 1. Four expressions for 118 as the sum of three positive integers have the same product: 118 is the smallest number that can be expressed as four sums with the same product in"
"this is not the case, and the smallest counterexample is the Mersenne number The evidence at hand suggests that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime values of appear to grow increasingly sparse as increases. For example, eight of the first 11 primes give rise to a Mersenne prime (the correct terms on Mersenne's original list), while is prime for only 43 of the first two million prime numbers (up to 32,452,843). The lack of any simple test to determine whether a given"
"Super-prime Super-prime numbers (also known as higher-order primes or prime-indexed primes) are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence begins That is, if ""p""(""i"") denotes the ""i""th prime number, the numbers in this sequence are those of the form ""p""(""p""(""i"")). used a computer-aided proof (based on calculations involving the subset sum problem) to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers. Their proof relies on a result resembling Bertrand's postulate, stating that (after the larger gap between super-primes 5 and"
"for other kinds of numbers. The record is currently held by 2 − 1 with 23,249,425 digits, found by GIMPS in December 2017. Its value is: The first and last 120 digits are shown above. The Great Internet Mersenne Prime Search (GIMPS) currently offers a US$3000 research discovery award for participants who download and run their free software and whose computer discovers a new Mersenne prime having fewer than 100 million digits. There are several prizes offered by the Electronic Frontier Foundation for record primes. GIMPS is also coordinating its long-range search efforts for primes of 100 million digits and"
"This definition is therefore equivalent to a definition as a prime number of the form for some prime . More generally, numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that be prime. The smallest composite Mersenne number with prime exponent ""n"" is . Mersenne primes are also noteworthy due to their connection to perfect numbers. A new Mersenne prime was found in December 2017. , 50 are now known. The largest known prime number is a Mersenne prime. Since 1997, all newly found Mersenne"
"second smallest to be both triangular and pentagonal (the third is 40755). It is also a idoneal number, a pentatope number, a pronic number, a Harshad number, and an untouchable number. 210 is also the first 71-gonal number, preceding 418. It is the first primorial number greater than 2 which is not adjacent to 2 primes (211 is prime, but 209 is not). It is the largest number ""n"" such that all primes between ""n""/2 and ""n"" yield a representation as a sum of two primes. 210 (number) 210 (two hundred [and] ten) is the natural number following 209 and"
"Primorial prime In mathematics, a primorial prime is a prime number of the form ""p""# ± 1, where ""p""# is the primorial of ""p"" (the product of the first ""n"" primes). Primality tests show that The first term of the second sequence is 0, because ""p""# = 1 is the empty product, and thus ""p""# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, as ""p""# = 2, and 2 − 1 = 1 is not prime. The first few primorial primes are , the largest known primorial prime is 1098133#"
"""proven"" unique prime was (10 + 1)/10001 or, using the notation above, (99990000)+ 1. It has 1128 digits. The record has been improved many times since then. the largest proven unique prime is formula_45, it has 20160 digits. The first unique primes in binary (base 2) are: The period length of them are: They include Fermat primes (the period length is a power of 2), Mersenne primes (the period length is a prime) and Wagstaff primes (the period length is twice an odd prime). Additionally, if ""n"" is a natural number which is not equal to 1 or 6, than"
"The prime is . Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered. On January 25, 2013, Curtis Cooper, a mathematician at the University of Central Missouri, discovered a 48th Mersenne prime, (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network. On January 19, 2016, Cooper published his discovery of a 49th Mersenne prime, (a number with 22,338,618 digits), as a result of a search executed by a GIMPS server network. This was"
"Megaprime A megaprime is a prime number with at least one million decimal digits (whereas titanic prime is a prime number with at least 1,000 digits, and gigantic prime has at least 10,000 digits). , 325 (probable) megaprimes are known, including 309 definite primes and 16 probable primes. The first to be found was the Mersenne prime 2−1 with 2,098,960 digits, discovered in 1999 by Nayan Hajratwala, a participant in the distributed computing project GIMPS. The term bevaprime has been proposed as a term for a prime with at least 1,000,000,000 digits. In fact, ""almost all"" primes are megaprimes, as"
"leads to a different but equivalent definition of the primes: they are the numbers with exactly two positive divisors, 1 and the number itself. Yet another way to say the same thing is that a number formula_1 is prime if it is greater than one and if none of the numbers formula_10 divides formula_1 evenly. The first 25 prime numbers (all the prime numbers less than 100) are: No even number formula_1 greater than 2 is prime because any such number can be expressed as the product formula_13. Therefore, every prime number other than 2 is an odd number, and"
"for numbers of special forms, such as Mersenne numbers. the largest known prime number has 23,249,425 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm."
"the prime numbers from 2 to 97 are: where ? indicates that the position (or whether it exists at all) is unknown as of 2012. A related sequence of numbers determined by the largest prime factor of one plus the product of the previous numbers (rather than the smallest prime factor) is also known as the Euclid–Mullin sequence. It grows more quickly, but is not monotonic. The numbers in this sequence are Not every prime number appears in this sequence, and the sequence of missing primes, has been proven to be infinite. It is also possible to generate modified versions"
"The complete list: The largest is the 8-digit 73939133. All primes above 5 end with digit 1, 3, 7 or 9, so a right-truncatable prime can only contain those digits after the leading digit. There are 15 primes which are both left-truncatable and right-truncatable. They have been called two-sided primes. The complete list: A left-truncatable prime is called restricted if all of its left extensions are composite i.e. there is no other left-truncatable prime of which this prime is the left-truncated ""tail"". Thus 7937 is a restricted left-truncatable prime because the nine 5-digit numbers ending in 7937 are all composite,"
"""F"" is prime for 8 of the first 10 primes ""p""; the exceptions are ""F"" = 1 and ""F"" = 4181 = 37 × 113. However, Fibonacci primes appear to become rarer as the index increases. ""F"" is prime for only 26 of the 1,229 primes ""p"" below 10,000. The number of prime factors in the Fibonacci numbers with prime index are: , the largest known certain Fibonacci prime is ""F"", with 21925 digits. It was proved prime by Mathew Steine and Bouk de Water in 2015. The largest known probable Fibonacci prime is ""F"". It was found by Henri"
"Gigantic prime A gigantic prime is a prime number with at least 10,000 decimal digits. The term appeared in ""Journal of Recreational Mathematics"" in the article ""Collecting gigantic and titanic primes"" (1992) by Samuel Yates. Chris Caldwell, who continued Yates' collection in The Prime Pages, reports that he changed the requirement from Yates' original 5,000 digits to 10,000 digits, when he was asked to revise the article after the death of Yates. Few primes of that size were known then, but a modern personal computer can find many in a day. The first discovered gigantic prime was the Mersenne prime"
"long and may be the largest known dihedral prime . Dihedral prime A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation (normally or upside down), and surface (actual display or reflection on a mirror). The first few decimal dihedral primes are The smallest dihedral prime that reads differently with each orientation and surface combination is 120121 which becomes 121021 (upside down), 151051 (mirrored), and 150151 (both upside down and mirrored). The digits 0, 1 and 8 remain the same regardless"
"the progression of the largest known prime number in ascending order. Here M = 2 − 1 is the Mersenne number with exponent ""n"". The longest record-holder known was M = 524,287, which was the largest known prime for 144 years. No records are known before 1456. GIMPS found the fourteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world. A list of the 5,000 largest known primes is maintained by Chris K. Caldwell, of which the twenty largest are listed below. Largest known prime number The largest known prime number () is"
"the smallest prime number, and the only even prime number (for this reason it is sometimes called ""the oddest prime""). The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, and the first Ramanujan prime. Two is the third (or fourth) Fibonacci number. Two is the base of the binary system, the numeral system with the least number of tokens allowing to denote a natural number substantially more concise ( tokens), compared to a direct representation by the corresponding count"
"a Proth number can be tested with Proth's theorem, which states that a Proth number formula_6 is prime if and only if there exists an integer formula_7 for which The largest known Proth prime is formula_9, and is 9,383,761 digits long. It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016. It is also the largest known non-Mersenne prime. Proth number In number theory, a Proth number is a number of the form where formula_2 is an odd positive integer and formula_3 is a positive integer such that formula_4. They are"
"A068188). The largest known tetradic prime is where formula_2 is a repunit, that is, a number which contains only the digit 1 repeated formula_3 times. The prime has 180,055 decimal digits. Tetradic number A tetradic number, also known as a four-way number, is a number that remains the same when flipped back to front, flipped front to back, mirrored up-down, or flipped up-down. The only numbers that remain the same which turned up-side-down or mirrored are 0, 1, and 8, so a tetradic number is a palindromic number containing only 0, 1, and 8 as digits. The first few tetradic"
"Supersingular prime (moonshine theory) In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group ""M"", which is the largest sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31), as well as 41, 47, 59, and 71. The non-supersingular primes are 37, 43, 53, 61, 67, and any prime number greater than or equal to 73. All supersingular primes are Chen primes, but 37, 53, and 67 are also Chen primes, and"
"example, the 29th Mersenne prime was discovered ""after"" the 30th and the 31st. Similarly, was followed by two smaller Mersenne primes, first 2 weeks later and then 8 months later. was the first discovered prime number with more than 10 million decimal digits. The largest known Mersenne prime is also the largest known prime number. In modern times, the largest known prime has almost always been a Mersenne prime. Since they are prime numbers, Mersenne primes are divisible only by 1 and by themselves. However, not all Mersenne numbers are Mersenne primes, and the composite Mersenne numbers may be factored"
"five-digit repunit 11111, and the largest prime number for which the decimal period of its multiplicative inverse is 5: 271 (number) 271 (two hundred [and] seventy-one) is the natural number after and before . 271 is a twin prime with 269, a cuban prime (a prime number that is the difference of two consecutive cubes), and a centered hexagonal number. It is the smallest prime number bracketed on both sides by numbers divisible by cubes, and the smallest prime number bracketed by numbers with five primes (counting repetitions) in their factorizations: After 7, 271 is the second-smallest Eisenstein–Mersenne prime, one"