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Table Of Prime Factors
The tables contain the prime factorization of the natural numbers from 1 to 1000. When ''n'' is a prime number, the prime factorization is just ''n'' itself, written in bold below. The number 1 is called a unit. It has no prime factors and is neither prime nor composite. Properties Many properties of a natural number ''n'' can be seen or directly computed from the prime factorization of ''n''. *The multiplicity of a prime factor ''p'' of ''n'' is the largest exponent ''m'' for which ''pm'' divides ''n''. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since ''p'' = ''p''1). The multiplicity of a prime which does not divide ''n'' may be called 0 or may be considered undefined. *Ω(''n''), the big Omega function, is the number of prime factors of ''n'' counted with multiplicity (so it is the sum of all prime factor multiplicities). *A prime number has Ω(''n'') = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number ( RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 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, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … . The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition. Properties Algebraic properties 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''''n'' (i.e. the group of units of the ring Z/''p''''n''Z) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''. A ''decimal numeral'' (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in or ). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in " is the approximation of to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value. The numbers that may be represented in the decimal system are the #Decimal fractions, decimal fractions. That is, fraction (mathematics), fract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frugal Number
In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 125 = 53, 128 = 27, 243 = 35, and 256 = 28 are frugal numbers . The first frugal number which is not a prime power is 1029 = 3 × 73. In base 2, thirty-two is a frugal number, since 32 = 25 is written in base 2 as 100000 = 10101. The term economical number has been used for a frugal number, but also for a number which is either frugal or equidigital. Mathematical definition Let b > 1 be a number base, and let K_b(n) = \lfloor \log_b \rfloor + 1 be the number of digits in a natural number n for base b. A natural number n has the prime factorisation : n = \prod_ p^ where v_p(n) is the ''p''-adic valuation of n, and n is an frugal number in base b if : K_b(n) > \sum_ K_b(p) + \sum_ K_b(v_p(n)). See also *Equidigital number In number theory, an equidigital n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Number
Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 602 = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are regular. These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study. * In number theory, these numbers are called 5-smooth, because they can be characterized as having only 2, 3, or 5 as their prime factors. This is a specific case of the more general - smooth numbers, the numbers that have no prime factor greater * In the study of Babylonian mathematics, the divisors of powers of 60 are called regular numbers or regular sexagesimal numbers, and are of great importance in this area because of the sexagesimal (base 60) number system that the Babylonians used for writing their numbers, and that was ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Number
In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers. Definition A positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310. The first five primorials are: : 2, 6, 30, 210, 2310 . The sequence also includes as empty product. Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Function
In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer variable ''n'' such that whenever ''a'' and ''b'' are coprime, the function applied to the product ''ab'' is the sum of the values of the function applied to ''a'' and ''b'':Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207online/ref> f(a b) = f(a) + f(b). Completely additive An additive function ''f''(''n'') is said to be completely additive if f(a b) = f(a) + f(b) holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If ''f'' is a completely additive function then ''f''(1) = 0. Every completely additive function is additive, but not vice versa. Examples Examples of arithmetic functions which are completely additive are: * The restriction of the logarit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphenic Number
In number theory, a sphenic number (from grc, σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. Definition A sphenic number is a product ''pqr'' where ''p'', ''q'', and ''r'' are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes. Examples The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are : 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... the largest known sphenic number is :(282,589,933 − 1) × (277,232,917 − 1) × (274,207,281 − 1). It is the product of the three largest known primes. Divisors All sphenic numbers have exactly eight divisors. If we express the sphenic number as n = p \cdot q \cdot r, where ''p'', ''q'', and ''r'' are distinct primes, then the set of divisors of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted . Definition For any positive integer , define as the sum of the primitive th roots of unity. It has values in depending on the factorization of into prime factors: * if is a square-free positive integer with an even number of prime factors. * if is a square-free positive integer with an odd number of prime factors. * if has a squared prime factor. The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where is the Kronecker delta, is the Liouville function, is the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
