"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"
"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#"
"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"
"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"
"Prime gap A prime gap is the difference between two successive prime numbers. The ""n""-th prime gap, denoted ""g"" or ""g""(""p"") is the difference between the (""n"" + 1)-th and the ""n""-th prime numbers, i.e. We have ""g"" = 1, ""g"" = ""g"" = 2, and ""g"" = 4. The sequence (""g"") of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered. The first 60 prime gaps are: By the definition of ""g"" every prime can be written as The first, smallest, and only odd prime gap is the gap of size 1 between 2, the"
"""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"
"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"
"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"
"101 (number) 101 (one hundred [and] one) is the natural number following 100 and preceding 102. It is variously pronounced ""one hundred and one"" / ""a hundred and one"", ""one hundred one"" / ""a hundred one"", and ""one oh one"". As an ordinal number, 101st (one hundred [and] first), rather than 101th, is the correct form. 101 is: Given 101, the Mertens function returns 0. It is the second prime having this property. For a 3-digit number in base 10, this number has a relatively simple divisibility test. The candidate number is split into groups of four, starting with the"
"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"
"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"
"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"
"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"
"that divides the smallest composite Euclid number. In this case 59 divides the Euclid number 13# + 1 = 2 × 3 × 5 × 7 × 11 × 13 + 1 = 59 × 509 = 30031. 59 is the highest integer a single symbol may represent in the Sexagesimal system. Fifty-nine is: 59 (number) 59 (fifty-nine) is the natural number following 58 and preceding 60. Fifty-nine is the 17th prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an"
"achieve this number of primes is Primeval numbers can be composite. The first is 1037 = 17×61. A Primeval prime is a primeval number which is also a prime number: The following table shows the first seven primeval numbers with the obtainable primes and the number of them. In base 12, the primeval numbers are: (using inverted two and three for ten and eleven, respectively) The number of primes that can be obtained from the primeval numbers is: (written in base 10) Note that 13, 115 and 135 are composite: 13 = 3×5, 115 = 7×1Ɛ, and 135 = 5×31."
"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"
"100 (number) 100 or one hundred (Roman numeral: Ⅽ) is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of ""hundred"" to describe the long hundred of six score or 120. 100 is the square of 10 (in scientific notation it is written as 10). The standard SI prefix for a hundred is ""hecto-"". 100 is the basis of percentages (""per cent"" meaning ""per hundred"" in Latin), with 100% being a full amount. 100 is the sum of"
"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."
"asymptotic density. The upper density of the happy numbers is greater than 0.18577, and the lower density is less than 0.1138. The number of happy numbers up to 10 for ""n"" = 1 to 20 is A happy prime is a number that is both happy and prime. The happy primes below 500 are All numbers, and therefore all primes, of the form 10 + 3 or 10 + 9 for ""n"" greater than 0 are happy. This does not mean that these are the only happy primes, as evidenced by the sequence above. To see this, note that The"
"""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"
"121 (number) 121 (one hundred [and] twenty-one) is the natural number following 120 and preceding 122. One hundred [and] twenty-one is a square and is the sum of three consecutive primes (37 + 41 + 43). There are no squares besides 121 known to be of the form formula_1, where ""p"" is prime (3, in this case). Other such squares must have at least 35 digits. There are only two other squares known to be of the form n! + 1, supporting Brocard's conjecture. Another example of 121 being of the few examples supporting a conjecture is that Fermat conjectured"
"numbers past one trillion in the short scale system, in ascending powers of 1000, are as follows: quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion, tredecillion, quattuordecillion, quindecillion, sexdecillion, septendecillion, octodecillion, novemdecillion and vigintillion (which is 10 to the 63rd power, or a one followed by 63 zeros). The highest number in this series listed in modern dictionaries is centillion, which is 10 to the 303rd power. The interim powers of one thousand between vigintillion and centillion do not have standardized names, nor do any higher powers, but there are many ad hoc extensions in use. The highest number"
"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"
"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"
"Prime power In mathematics, a prime power is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The number 1 is not counted as a prime power. The prime powers are: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191,"
"first four perfect numbers are generated by the formula 2(2 − 1), with ""p"" a prime number, as follows: Prime numbers of the form 2 − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2 − 1 to be prime, it is necessary that ""p"" itself be prime. However, not all numbers of the form 2 − 1 with a prime ""p"" are prime; for example, 2 − 1 = 2047 = 23 × 89 is not a prime number. In fact, Mersenne primes are very rare—of the"
"prime number for 75 years, and the largest ever found by hand. was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876. Without finding a factor, Lucas demonstrated that is actually composite. No factor was found until a famous talk by Frank Nelson Cole in 1903. Without speaking a word, he went to a blackboard and raised 2 to the"
"function. The function is neither multiplicative nor additive. Prime gap A prime gap is the difference between two successive prime numbers. The ""n""-th prime gap, denoted ""g"" or ""g""(""p"") is the difference between the (""n"" + 1)-th and the ""n""-th prime numbers, i.e. We have ""g"" = 1, ""g"" = ""g"" = 2, and ""g"" = 4. The sequence (""g"") of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered. The first 60 prime gaps are: By the definition of ""g"" every prime can be written as The first, smallest, and only odd prime gap is"
"story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with Asperger syndrome. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel ""The Solitude of Prime Numbers"", in which they are portrayed as ""outsiders"" among integers. 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 probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (). In other words, the average gap between consecutive prime numbers among the first integers is roughly . Let be the prime-counting"
"palindromic prime 10 + 7426247 + 1 is also a happy prime with digits because the many 0s do not contribute to the sum of squared digits, and formula_13, which is a happy number. Paul Jobling discovered the prime in 2005. , the largest known happy prime is formula_14 (Mersenne prime). Its decimal expansion has digits. Unlike happy numbers, rearranging the digits of a happy prime will not necessarily create another happy prime. For instance, while 19 is a happy prime, 91 is not prime. The definition of happy numbers depends on the decimal (i.e., base 10) representation of the"
"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,"
"3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, ... . proved that this sequence contains only ones and prime numbers. However, it does not contain all the prime numbers, since the terms gcd(""n""+1, ""a"") are always odd and so never equal to 2. 587 is the smallest prime not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper it was conjectured to contain all (odd) primes, even though it is rather inefficient. Note"