"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"
"numbers are {1, 3, 5, 7, ...}. Any even number ""m"" has the form where ""k"" is again an integer. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}. A prime number is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than 2000 years and have led to many questions, only some of"
"5 5 (five) is a number, numeral, and glyph. It is the natural number following 4 and preceding 6. Five is the third prime number. Because it can be written as , five is classified as a Fermat prime; therefore a regular polygon with 5 sides (a regular pentagon) is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, and the third Mersenne prime exponent. Five is the first Wilson prime and the third factorial prime, also an alternating factorial. Five is the first good prime. It is"
"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"
"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"
"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."
"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"
"Fibonacci prime A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime. The first Fibonacci primes are : It is not known whether there are infinitely many Fibonacci primes. With the indexing starting with , the first 34 are ""F"" for the ""n"" values : In addition to these proven Fibonacci primes, there have been found probable primes for Except for the case ""n"" = 4, all Fibonacci primes have a prime index, because if ""a"" divides ""b"", then formula_1 also divides formula_2, but not every prime is the index of a Fibonacci prime."
"Prime triplet In mathematics, 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 have the form (""p"", ""p"" + 2, ""p"" + 6) or (""p"", ""p"" + 4, ""p"" + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself). The first prime triplets are A prime triplet contains"
"""p"" = 6521953289619 × 2 − 5. Prime triplet In mathematics, 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 have the form (""p"", ""p"" + 2, ""p"" + 6) or (""p"", ""p"" + 4, ""p"" + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself). The first"
"prime is a centered heptagonal number that is prime. The first few centered heptagonal primes are and centered heptagonal twin prime numbers are Centered heptagonal number A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. The centered heptagonal number for ""n"" is given by the formula This can also be calculated by multiplying the triangular number for (""n"" – 1) by 7, then adding 1. The first few centered heptagonal numbers are 1, 8, 22, 43, 71, 106,"
"in a generalized way like prime numbers include prime elements and prime ideals. A natural number (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it. The numbers greater than 1 that are not prime are called composite numbers. In other words, formula_1 is prime if formula_1 items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange formula_1 dots into"
"but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). For a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences. Other examples of sequences include"
"Primeval number In mathematics, a primeval number is a natural number ""n"" for which 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 the same way for any smaller natural number. Primeval numbers were first described by Mike Keith. The first few primeval numbers are The number of primes that can be obtained from the primeval numbers is The largest number of primes that can be obtained from a primeval number with ""n"" digits is The smallest ""n""-digit number to"
"Primeval number In mathematics, a primeval number is a natural number ""n"" for which 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 the same way for any smaller natural number. Primeval numbers were first described by Mike Keith. The first few primeval numbers are The number of primes that can be obtained from the primeval numbers is The largest number of primes that can be obtained from a primeval number with ""n"" digits is The smallest ""n""-digit number to"
"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#"
"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 quadruplet A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {""p"", ""p""+2, ""p""+6, ""p""+8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. The first eight prime quadruplets are: All prime quadruplets except {5, 7, 11, 13} are of the form {30""n"" + 11, 30""n"" + 13, 30""n"" + 17, 30""n"" + 19} for some integer ""n"". (This structure is necessary to ensure that none of the four primes is divisible by 2, 3 or 5). A prime quadruplet of"
"Ramanujan prime In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function. In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is: where formula_2 is the prime-counting function, equal to the number of primes less than or equal to ""x"". The converse of this result is the definition of Ramanujan primes: The first five Ramanujan primes are thus 2, 11, 17, 29, and"
"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"
"Permutable prime 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 and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes. In base 10, all the permutable primes with fewer than 49,081 digits are known Of the above, there are 16 unique permutation sets, with smallest elements Note R = formula_1 is a repunit, a number consisting only of ""n"" ones (in base"
"a rectangular grid that is more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. and are both composite. The divisors of a natural number formula_1 are the numbers that divide formula_1 evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This idea"
"character usually has a descender, as, for example, in . Five can refer to: 5 5 (five) is a number, numeral, and glyph. It is the natural number following 4 and preceding 6. Five is the third prime number. Because it can be written as , five is classified as a Fermat prime; therefore a regular polygon with 5 sides (a regular pentagon) is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, and the third Mersenne prime exponent. Five is the first Wilson prime and the third"
"193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, ... The prime powers are those positive integers that are divisible by exactly one prime number; prime powers and related concepts are also called primary numbers, as in the primary decomposition. Prime powers are powers of prime numbers. Every prime power (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo ""p"" (or equivalently, the group of units of the ring Z/""p""Z) is cyclic. The number of elements of a finite field is always a prime power and conversely, every prime power"
"Factorial prime A factorial prime is a prime number that is one less or one more than a factorial (all factorials > 1 are even). The first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : ""n""! − 1 is prime for : ""n""! + 1 is prime for : No other factorial primes are known . Absence of primes to both sides of a factorial ""n""! implies a run of at least 2""n""+1 consecutive composite numbers, since ""n""! ± ""k"" is divisible by ""k"" for 2 ≤ ""k"" ≤ ""n"". However,"
"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"
"is a list of the 5 largest known generalized Fermat primes. They are all megaprimes. the whole top-5 was discovered by participants in the PrimeGrid project. On the Prime Pages you can perform a search yielding the current top 100 generalized Fermat primes. Fermat number In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where ""n"" is a nonnegative integer. The first few Fermat numbers are: If 2 + 1 is prime, and ""k"" > 0, it can be shown that ""k"" must be a power of two."
"593 (number) 593 (five hundred [and] ninety-three) is the natural number following 592 and preceding 594. 593 is an odd number. It is a prime number, an example of what Paul Erdős and Ernst G. Straus called a Good prime, or a prime whose square is greater than the product of its neighboring two primes. As such it is part of sequence at the On-Line Encyclopedia of Integer Sequences. Justin Smith calls 593 a right prime because it remains prime after dropping any number of digits from the right: 593, 59, and 5 are all prime. It is a Sophie"
"the only prime number which is the sum of four consecutive primes (2,3,5,7). Any other four consecutive primes summed would always produce an even number, thereby divisible by 2 and so not prime. In Catalan, 17 is the first compound number (""disset""). The numbers 11 (""onze"") through 16 (""setze"") have their own names. In French, 17 is the first compound number (""dix-sept""). The numbers 11 (""onze"") through 16 (""seize"") have their own names. In Italian, 17 is also the first compound number (""diciassette""), whereas sixteen is ""sedici"". Seventeen is: 17 (number) 17 (seventeen) is the natural number following 16 and"
"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"
"18. Adding 19, however, gives 510529, which is prime. Hence 19 is a Fortunate number. The Fortunate number for ""p""# is always above ""p"" and all its divisors are larger than ""p"". This is because ""p""#, and thus ""p""# + ""m"", is divisible by the prime factors of ""m"" not larger than ""p"". The Fortunate numbers for the first primorials are: The Fortunate numbers sorted in numerical order with duplicates removed: Reo Fortune conjectured that no Fortunate number is composite (""Fortune's conjecture""). A Fortunate prime is a Fortunate number which is also a prime number. , all the known Fortunate"
"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"