In mathematics, a perfect power is a positive integer that can be resolved into equal factors, whose root can be extracted. I.e. a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3 n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are considered perfect powers. A sequence of perfect powers can be generated by iterating through the possible values for k; the first few ascending perfect powers in numerical order are: 2 2 = 4, 2 3 = 8, 3 2 = 9, 2 4 = 16, 4 2 = 16, 5 2 = 25, 3 3 = 27, 2 5 = 32, 6 2 = 36, 7 2 = 49, 2 6 = 64, 4 3 = 64, 8 2 = 64, … The sum of the reciprocals of the perfect powers is 1: ∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = 1. Which can be proved as follows: ∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 ∑ k = 0 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 = ∑ m = 2 ∞ 1 m = ∑ m = 2 ∞ = 1.
The first perfect powers without duplicates are:, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024... The sum of the reciprocals of the perfect powers p without duplicates is: ∑ p 1 p = ∑ k = 2 ∞ μ ≈ 0.874464368 … where μ is the Möbius function and ζ is the Riemann zeta function. According to Euler, Goldbach showed that the sum of 1/p − 1 over the set of perfect powers p, excluding 1 and excluding duplicates, is 1: ∑ p 1 p − 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + ⋯ = 1; this is sometimes known as the Goldbach–Euler theorem. Detecting whether or not a given natural number n is a perfect power may be accomplished in many different ways, with varying levels of complexity. One of the simplest such methods is to consider all possible values for k across each of the divisors of n, up to k ≤ log 2 n. So if the divisors of n are n 1, n 2, …, n j one of the values n 1 2, n 2 2
In mathematics, a square-free integer is an integer, divisible by no perfect square other than 1. That is, its prime factorization has one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32; the smallest positive square-free numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39... The radical of an integer is its largest square-free factor. An integer is only if it is equal to its radical. Any arbitrary positive integer n can be represented in a unique way as the product of a powerful number and a square-free integer, which are coprime; the square-free factor, called the square-free part of the number, is the largest square-free divisor k of n, coprime with n/k. The square-free part of an integer may be smaller than the largest square-free divisor. Any arbitrary positive integer n can be represented in a unique way as the product of a square and a square-free integer: n = m 2 k In this factorization, m is the largest divisor of n such that m2 is a divisor of n.
In summary, there are three square-free factors that are associated to every integer: the square-free part, the above factor k, the largest square-free factor. Each is a factor of the next one. All are deduced from a prime factorization: if n = ∏ i = 1 h p i e i is the prime factorization of n, where p 1, …, p h are distinct prime numbers the square-free part is ∏ e i = 1 p i, The square-free factor such the quotient is a square is ∏ e i odd p i, the largest square-free factor is ∏ i = 1 h p i. For example, if n = 75600 = 2 4 ⋅ 3 3 ⋅ 5 2 ⋅ 7, the square-free part is 7, the square-free factor such that the quotient is a square is 3 ⋅ 7 = 21, the largest square-free factor is 2 ⋅ 3 ⋅ 5 ⋅ 7 = 210. No algorithm is known for computing any of these square-free factors, faster than computing the complete prime factorization. In particular, there is no known polynomial-time algorithm for computing the square-free part of an integer, no known polynomial-time algorithm for determining whether a number is square-free.
On the opposite, polynomial-time algorithms are known for primality testing. This is a major difference between the arithmetic of the integers, the arithmetic of the univariate polynomials, as polynomial-time algorithms are known for square-free factorization of polynomials. A positive integer n is square-free if and only if in the prime factorization of n, no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor p of n, the prime p does not evenly divide n / p. N is square-free if and only if in every factorization n = ab, the factors a and b are coprime. An immediate result of this definition is. A positive integer n is square-free if and only if all abelian groups of order n are isomorphic, the case if and only if any such group is cyclic; this follows from the classification of finitely generated abelian groups. A integer n is square-free; this follows from the Chinese remainder theorem and the fact that a ring of the form Z / kZ is a field if and only if k is a prime.
For every positive integer n, the set of all positive divisors of n becomes a ordered set if we use divisibility as the order relation. This ordered set is always a distributive lattice, it is a Boolean algebra if and only. A positive integer n is square-free; the absolute value of the Möbius function is the indicator function for the square-free integers – that is, |μ| is equal to 1 if n is square-free, 0 if it is not. The Dirichlet series of this indicator function is ∑ n = 1 ∞ | μ | n s = ζ ζ, where ζ is the Riemann zeta function; this follows from the Euler product ζ ζ =
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, this representation is unique, up to the order of the factors. For example, 1200 = 24 × 31 × 52 = 2 × 2 × 2 × 2 × 3 × 5 × 5 = 5 × 2 × 5 × 2 × 3 × 2 × 2 =... The theorem says two things for this example: first, that 1200 can be represented as a product of primes, second, that no matter how this is done, there will always be four 2s, one 3, two 5s, no other primes in the product; the requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique. This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime factorization into primes would not be unique. Book VII, propositions 30, 31 and 32, Book IX, proposition 14 of Euclid's Elements are the statement and proof of the fundamental theorem.
If two numbers by multiplying one another make some number, any prime number measure the product, it will measure one of the original numbers. Proposition 30 is referred to as Euclid's lemma, it is the key in the proof of the fundamental theorem of arithmetic. Any composite number is measured by some prime number. Proposition 31 is proved directly by infinite descent. Any number either is measured by some prime number. Proposition 32 is derived from proposition 31, proves that the decomposition is possible. If a number be the least, measured by prime numbers, it will not be measured by any other prime number except those measuring it. Book IX, proposition 14 is derived from Book VII, proposition 30, proves that the decomposition is unique – a point critically noted by André Weil. Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.
Every positive integer n > 1 can be represented in one way as a product of prime powers: n = p 1 n 1 p 2 n 2 ⋯ p k n k = ∏ i = 1 k p i n i where p1 < p2 <... < pk are primes and the ni are positive integers. This representation is extended to all positive integers, including 1, by the convention that the empty product is equal to 1; this representation is called the canonical representation of n, or the standard form of n. For example, 999 = 33×37, 1000 = 23×53, 1001 = 7×11×13. Note that factors p0 = 1 may be inserted without changing the value of n. In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers: n = 2 n 1 3 n 2 5 n 3 7 n 4 ⋯ = ∏ i = 1 ∞ p i n i, where a finite number of the ni are positive integers, the rest are zero. Allowing negative exponents provides a canonical form for positive rational numbers; the canonical representations of the product, greatest common divisor, least common multiple of two numbers a and b can be expressed in terms of the canonical representations of a and b themselves: a ⋅ b = 2 a 1 + b 1 3 a 2 + b 2 5 a 3 + b 3 7 a 4 + b 4 ⋯ = ∏ p i a i + b i, gcd = 2 min 3 min ( a 2, b 2
An Achilles number is a number, powerful but not a perfect power. A positive integer n is a powerful number if, for every prime factor p of n, p2 is a divisor. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as mk, where m and k are positive integers greater than 1. Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, powerful but imperfect. Strong Achilles numbers are Achilles numbers whose Euler totients are Achilles numbers. A number n = p1a1p2a2…pkak is powerful if min ≥ 2. If in addition gcd = 1 the number is an Achilles number; the Achilles numbers up to 5000 are: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000. The smallest pair of consecutive Achilles numbers is: 5425069447 = 73 × 412 × 972 5425069448 = 23 × 260412 108 is a powerful number.
Its prime factorization is 22 · 33, thus its prime factors are 2 and 3. Both 22 = 4 and 32 = 9 are divisors of 108. However, 108 cannot be represented as mk, where m and k are positive integers greater than 1, so 108 is an Achilles number. 360 is not an Achilles number. One of its prime factors is 5 but 360 is not divisible by 52 = 25. 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but 22 = 4 and 72 = 49 are divisors of it. Nonetheless, it is a perfect power: 784 = 2 4 ⋅ 7 2 = 2 ⋅ 7 2 = 2 = 28 2. So it is not an Achilles number. 500 = 22 × 53 is a strong Achilles number as its Euler totient of 200 = 23 × 52 is an Achilles number
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, 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 a prime itself or can be factorized as a product of primes, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 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. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number 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, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.
Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.
The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number, its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance; the number 12 has an abundance of 4, for example. A number n for which the sum of divisors σ>2n, or, the sum of proper divisors s>n. Abundance is the value σ-2n; the first 28 abundant numbers are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, …. For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12, whose sum is 36; because 36 is more than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12; the smallest odd abundant number is 945. The smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes.
If A represents the smallest abundant number not divisible by the first k primes for all ϵ > 0 we have: 2 − ϵ < ln A < 2 + ϵ for sufficiently large k. Infinitely many and odd abundant numbers exist; the set of abundant numbers has a non-zero natural density. Marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480. Every multiple of a perfect number is abundant. For example, every multiple of 6 is abundant because the divisors include 1, n/2, n/3, n/6 which sum to n + 1; every multiple of an abundant number is abundant. For example, every multiple of 20 is abundant because n/2 + n/4 + n/5 + n/10 + n/20 = n + n/10; every integer greater than 20161 can be written as the sum of two abundant numbers. An abundant number, not a semiperfect number is called a weird number. An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found. Numbers whose sum of proper factors equals the number itself are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers.
The first known classification of numbers as deficient, perfect or abundant was by Nicomachus in his Introductio Arithmetica, which described abundant numbers as like deformed animals with too many limbs. The abundancy index of n is the ratio σ/n. Distinct numbers n1, n2... with the same abundancy index are called friendly numbers. The sequence of least numbers n such that σ > kn, in which a2 = 12 corresponds to the first abundant number, grows quickly. The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 33 × 52 × 72 × 11 × 13 × 17 × 19 × 23 × 29. If p = is a list of primes p is termed abundant if some integer composed only of primes in p is abundant. A necessary and sufficient condition for this is that the product of pi/ be at least 2. Tattersall, James J.. Elementary Number Theory in Nine Chapters. Cambridge University Press. ISBN 978-0-521-85014-8. Zbl 1071.11002. The Prime Glossary: Abundant number Weisstein, Eric W. "Abundant Number". MathWorld. Abundant number at PlanetMath.org
Prentice Hall is a major educational publisher owned by Pearson plc. Prentice Hall publishes print and digital content for the higher-education market. Prentice Hall distributes its technical titles through the Safari Books Online e-reference service. On October 13, 1913, law professor Charles Gerstenberg and his student Richard Ettinger founded Prentice Hall. Gerstenberg and Ettinger took their mothers' maiden names—Prentice and Hall—to name their new company. Prentice Hall was acquired by Gulf+Western in 1984, became part of that company's publishing division Simon & Schuster. Publication of trade books ended in 1991. Simon & Schuster's educational division, including Prentice Hall, was sold to Pearson by G+W successor Viacom in 1998. There were two or more authors, their books turned up missing. One book'The Roof Builder's Handbook' is still being sold in 2018 for as much as $230 per new copy, but the author William C. McElroy was told by Pearson that all new books were either destroyed or went missing in 1995.
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