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Leyland Number
In number theory, a Leyland number is a number of the form :x^y + y^x where ''x'' and ''y'' are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are : 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124 . The requirement that ''x'' and ''y'' both be greater than 1 is important, since without it every positive integer would be a Leyland number of the form ''x''1 + 1''x''. Also, because of the commutative property of addition, the condition ''x'' ≥ ''y'' is usually added to avoid double-covering the set of Leyland numbers (so we have 1 < ''y'' ≤ ''x''). Leyland primes A Leyland prime is a Leyland number that is . The first such primes are: : 17,[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
118 (number)
118 (one hundred [and] eighteen) is the natural number following 117 (number), 117 and preceding 119 (number), 119. In mathematics There is no answer to the equation Euler's totient function, φ(''x'') = 118, making 118 a nontotient. Four expressions for 118 as the sum of three positive integers have the same product: :14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72 = 118 and :14 × 50 × 54 = 15 × 40 × 63 = 18 × 30 × 70 = 21 × 25 × 72 = 37800. 118 is the smallest number that can be expressed as four sums with the same product in this way. Because of its expression as , it is a Leyland number#Leyland_number_of_the_second_kind, Leyland number of the second kind. 118!! - 1 is a prime number, where !! denotes the double factorial (the product of even integers up to 118). See also * 118 (other) References [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1124 (number)
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000. A group of one thousand units is sometimes known, from Ancient Greek, as a chiliad. A period of one thousand years may be known as a chiliad or, more often from Latin, as a millennium. The number 1000 is also sometimes described as a short thousand in medieval contexts where it is necessary to distinguish the Germanic concept of 1200 as a long thousand. It is the first 4-digit integer. Notation * The decimal representation for one thousand is ** 1000—a one followed by three zeros, in the general notation; ** 1 × 103—in engineering notation, which for this number coincides with: ** 1 × 103 exactly—in scientific normalized exponential notation; ** 1 E+3 exactly—in scientific E notation. * The SI prefix for a thousand units is "kilo-", abbreviated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
431 (number)
400 (four hundred) is the natural number following 399 and preceding 401. Mathematical properties A circle is divided into 400 grads. Integers from 401 to 499 400s 401 401 is a prime number, tetranacci number, Chen prime, prime index prime * Eisenstein prime with no imaginary part * Sum of seven consecutive primes (43 + 47 + 53 + 59 + 61 + 67 + 71) * Sum of nine consecutive primes (29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61) * Mertens function returns 0, * Member of the Mian–Chowla sequence. 402 402 = 2 × 3 × 67, sphenic number, nontotient, Harshad number, number of graphs with 8 nodes and 9 edges * HTTP status code for "Payment Required". *The area code for Nebraska. 403 403 = 13 × 31, heptagonal number, Mertens function returns 0. * First number that is the product of an emirp pair. * HTTP 403, the status code for "Forbidden" * Also in the name of a retirement plan in the United States, 403(b). * The area code for southern Alberta. 404 40 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
300 (number)
300 (three hundred) is the natural number following 299 and preceding 301. In Mathematics 300 is a composite number and the 24th triangular number. It is also a second hexagonal number. Integers from 301 to 399 300s 301 302 303 304 305 306 307 308 309 310s 310 311 312 313 314 315 315 = 32 × 5 × 7 = D_ \!, rencontres number, highly composite odd number, having 12 divisors. It is a Harshad number, as it is divisible by the sum of its digits. It is a Zuckerman number, as it is divisible by the product of its digits. 316 316 = 22 × 79, a centered triangular number and a centered heptagonal number. 317 317 is a prime number, Eisenstein prime with no imaginary part, Chen prime, one of the rare primes to be both right and left-truncatable, and a strictly non-palindromic number. 317 is the exponent (and number of ones) in the fourth base-10 repunit prime. 318 319 319 = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
192 (number)
192 (one hundred ndninety-two) is the natural number following 191 and preceding 193. In mathematics 192 has the prime factorization 2^6\times 3. Because it has so many small prime factors, it is the smallest number with exactly 14 divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...s, namely 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96 and 192. Because its only prime factors are 2 and 3, it is a 3- smooth number. 192 is a Leyland number of the second kind using 2 & 8 (2^8-8^2). See also * 192 (other) References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
79 (number)
79 (seventy-nine) is the natural number following 78 (number), 78 and preceding 80 (number), 80. In mathematics 79 is: * An even and odd numbers, odd number. * The smallest number that can not be represented as a sum of fewer than 19 fourth powers. * The 22nd prime number (between and ) * An isolated prime without a twin prime, as 77 and 81 are composite. * The smallest prime number ''p'' for which the real quadratic field Q[] has Ideal class group, class number greater than 1 (namely 3). * A cousin prime with 83. * An emirp in base 10, because the reverse of 79, 97 (number), 97, is also a prime. * A Fortunate prime. * A circular prime. * A prime number that is also a Gaussian prime (since it is of the form ). * A happy prime. * A Higgs prime. * A lucky prime. * A permutable prime, with 97 (number), ninety-seven. * A Pillai prime, because 23Factorial, ! + 1 is divisible by 79, but 79 is not one more than a Multiple (mathematics), multiple of 23 (number), 23. * A regular prime. * A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
28 (number)
28 (twenty-eight) is the natural number following 27 (number), 27 and preceding 29 (number), 29. In mathematics Twenty-eight is a composite number and the second perfect number as it is the sum of its proper divisors: 1+2+4+7+14=28. As a perfect number, it is related to the Mersenne prime 7, since 2^\times (2^-1)=28. The next perfect number is 496 (number), 496, the previous being 6 (number), 6. Though perfect, 28 is not the aliquot sum of any other number other than itself; thus, it is not part of a multi-number aliquot sequence. Twenty-eight is the sum of the totient function for the first nine integers. Since the greatest prime factor of 28^+1=785 is 157, which is more than 28 twice, 28 is a Størmer number. Twenty-eight is a harmonic divisor number, a happy number, the 7th triangular number, a hexagonal number, a Leyland number#Leyland number of the second kind, Leyland number of the second kind (2^6-6^2), and a centered nonagonal number. It appears in the Padovan sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime number, prime, or the Unit (ring theory), unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n)—and more precisely, because of the failure of unique factorization in their rings of integers—that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. Definition For n \geq 1, let :\zeta_n=e^\in\C. This is a primitive nth root of unity. Then the nth cyclotomic field is the field extension \mathbb(\zeta_n) of \mathbb generated by \zeta_n. Properties * The nth cyclotomic polynomial :: \Phi_n(x) = \prod_\stackrel\!\!\! \left(x-e^\right) = \prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of \zeta_n over \Q. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probable Prime
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer ''n'', choose some integer ''a'' that is not a multiple of ''n''; (typically, we choose ''a'' in the range ). Calculate . If the result is not 1, then ''n'' is composite. If the result is 1, then ''n'' is likely to be prime; ''n'' is then called a probable prime to base ''a''. A weak probable prime to base ''a'' is an integer that is a probable prime to base ''a'', but which is not a strong probable prime to base ''a'' (see below). For a fixed base ''a'', it is unusual fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
