|
50,000
50,000 (fifty thousand) is the natural number that comes after 49,999 and before 50,001 . Selected numbers in the range 50001–59999 50001 to 50999 * 50069 = 11 + 22 + 33 + 44 + 55 + 66 * 50400 = highly composite number * 50625 = 154, smallest fourth power that can be expressed as the sum of only five distinct fourth powers, palindromic in base 14 (1464114) * 50653 = 373, palindromic in base 6 (10303016) 51000 to 51999 * 51076 = 2262, palindromic in base 15 (1020115) * 51641 = Markov number * 51984 = 2282 = 373 + 113. the smallest square to the sum of only five distinct fourth powers. 52000 to 52999 * 52488 = 3-smooth number * 52633 = Carmichael number 53000 to 53999 * 53016 = pentagonal pyramidal number * 53361 = 2312 sum of the cubes of the first 21 positive integers 54000 to 54999 * 54205 = Zeisel number * 54688 = 2-automorphic number * 54748 = narcissistic number * 54872 = 383, palindromic in base 9 (832389) * 54901 = chiliagonal number 55000 to 55999 * 55296 = 3-smooth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Numbers
A prime number (or a prime) is a natural number greater than 1 that is not a product of 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, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. 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 that is 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 \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superior Highly Composite Number
In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to ''some'' positive power of itself than any other number. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer. The first 10 superior highly composite numbers and their factorization are listed. For a superior highly composite number ''n'' there exists a positive real number ''ε'' such that for all natural numbers ''k'' smaller than ''n'' we have :\frac\geq\frac and for all natural numbers ''k'' larger than ''n'' we have :\frac>\frac where ''d(n)'', the divisor function, denotes the number of divisors of ''n''. The term was coined by Ramanujan (1915). For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12. \frac\approx 1.414, \frac=1.5, \frac\approx 1.633, \ ... [...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, resp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seventeen Or Bust
Seventeen or Bust was a volunteer computing project started in March 2002 to solve the last seventeen cases in the Sierpinski problem. The project solved eleven cases before a server loss in April 2016 forced it to cease operations. Work on the Sierpinski problem moved to PrimeGrid, which solved a twelfth case in October 2016. Five cases remain unsolved . Goals The goal of the project was to prove that 78557 is the smallest Sierpinski number, that is, the least odd ''k'' such that ''k''·2''n''+1 is composite (i.e. not prime) for all ''n'' > 0. When the project began, there were only seventeen values of ''k'' 0 (or else ''k'' has algebraic factorizations for some ''n'' values and a finite prime set that works only for the remaining ''n''). For example, for the smallest known Sierpinski number, 78557, the covering set is . For another known Sierpinski number, 271129, the covering set is . Each of the remaining sequences has been tested and none has a small cove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base 6
A senary () numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ... (also known as base-6, heximal, or seximal) has 6, six as its radix, base. It has been adopted independently by a small number of cultures. Like decimal, it is a semiprime, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to senary. In turn, the logic, senary logic refers to an extension of Jan Łukasiewicz's and Stephen Cole Kleene's ternary logic systems adjusted to explain the logic of statistical tests and missing data patterns in sciences using empirical methods. Formal definition The standard Set (mathematics), set of digits in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repunit Prime
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreations in the Theory of Numbers''. A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of March 2022, the largest known prime number , the largest probable prime ''R''8177207 and the largest elliptic curve primality prime ''R''49081 are all repunits. Definition The base-''b'' repunits are defined as (this ''b'' can be either positive or negative) :R_n^\equiv 1 + b + b^2 + \cdots + b^ = \qquad\mbox, b, \ge2, n\ge1. Thus, the number ''R''''n''(''b'') consists of ''n'' copies of the digit 1 in base-''b'' representation. The first two repunits base-''b'' for ''n'' = 1 and ''n'' = 2 are :R_1^ 1 \qquad \text \qquad R_2^ b+1\qquad\text\ , b, \ge2. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Divisor Number
In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are: : 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 . Examples For example, the harmonic divisor number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer: : \frac=2. The number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is: : \frac=5 5 is an integer, making 140 a harmonic divisor number. Factorization of the harmonic mean The harmonic mean of the divisors of any number can be expressed as the formula :H(n) = \frac where is the sum of th powers of the divisors of : is the number of divisors, and is the sum of divisors . All of the terms in this formula are multiplicative, but not completely multiplicative. Therefore, the harmonic mean is also multiplicative. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colossally Abundant Number
In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Formally, a number ''n'' is said to be colossally abundant if there is an ε > 0 such that for all ''k'' > 1, :\frac\geq\frac where ''σ'' denotes the sum-of-divisors function. All colossally abundant numbers are also superabundant numbers, but the converse is not true. The first 15 colossally abundant numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15 superior highly composite numbers, but neither set is a subset of the other. History Colossally abundant numbers were first studied by Ramanujan and his findings were intended to be included in his 1915 paper on highly composite numbers. Unfortunately, the publisher of the journal to which Ramanujan submitted his work, the London Mathematical Society, was in financ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carmichael Number
In number theory, a Carmichael number is a composite number n, which in modular arithmetic satisfies the congruence relation: :b^n\equiv b\pmod for all integers b. The relation may also be expressed in the form: :b^\equiv 1\pmod. for all integers b which are relatively prime to n. Carmichael numbers are named after American mathematician Robert Carmichael, the term having been introduced by Nicolaas Beeger in 1950 ( Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "''F'' numbers" for short). They are infinite in number. They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality. The Carmichael numbers form the subset ''K''1 of the Knödel numbers. Overview Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''b'', the number ''b'' − ''b'' is an integer multipl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
