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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 = 27th 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 * 53174 = number of partitions of 42 * 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 = chiliag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 , called trial division, tests whether is a multiple of any integer between 2 and . 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 produ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Superior Highly Composite Number
In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power. For any possible exponent, whichever integer has the greatest ratio is a superior highly composite 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 ten superior highly composite numbers and their factorization are listed. For a superior highly composite number there exists a positive real number such that for all natural numbers we have \frac\geq\frac where , the divisor function, denotes the number of divisors of . 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 composite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Seventeen Or Bust
PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client, others through the PRPNet client. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads. PrimeGrid awards badges to users in recognition of achieving certain defined levels of credit for work done. The badges have no intrinsic value but are valued by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Catalan Number
The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The -th Catalan number can be expressed directly in terms of the central binomial coefficients by :C_n = \frac = \frac \qquad\textn\ge 0. The first Catalan numbers for are : . Properties An alternative expression for is :C_n = - for n\ge 0\,, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a #Second proof, proof of the correctness of the formula. Another alternative expression is :C_n = \frac \,, which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the recurr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Base 6
A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like the decimal base 10, the base 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 the senary system. Formal definition The standard set of digits in the senary system is \mathcal_6 = \lbrace 0, 1, 2, 3, 4, 5\rbrace, with the linear order 0 < 1 < 2 < 3 < 4 < 5. Let be the of , where is the operation of string concatenation [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 October 2024, the largest known prime number , the largest probable prime ''R''8177207 and the largest elliptic curve primality-proven prime ''R''86453 are all repunits in various bases. 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^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Harmonic Divisor Number
In mathematics, a harmonic divisor number or Ore number 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 . Harmonic divisor numbers were introduced by Øystein Ore, who showed that every perfect number is a harmonic divisor number and conjectured that there are no odd harmonic divisor numbers other than 1. Examples The number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer: \frac=2. Thus 6 is a harmonic divisor number. Similarly, the number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is \frac=5. Since 5 is an integer, 140 is 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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Highly Composite Number
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(''N'') > ''d''(''n'') for all ''n'' < ''N''. For example, 6 is highly composite because ''d''(6)=4, and for ''n''=1,2,3,4,5, you get ''d''(''n'')=1,2,2,3,2, respectively, which are all less than 4. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. Ramanujan wrote a paper on highly composite numbers in 1915. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Markov Number
A Markov number or Markoff number is a positive integer ''x'', ''y'' or ''z'' that is part of a solution to the Markov Diophantine equation :x^2 + y^2 + z^2 = 3xyz,\, studied by . The first few Markov numbers are :1 (number), 1, 2 (number), 2, 5 (number), 5, 13 (number), 13, 29 (number), 29, 34 (number), 34, 89 (number), 89, 169 (number), 169, 194 (number), 194, 233 (number), 233, 433, 610, 985, 1325, ... appearing as coordinates of the Markov triples :(1, 1, 1), (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), (1, 13, 34), (1, 34, 89), (2, 29, 169), (5, 13, 194), (1, 89, 233), (5, 29, 433), (1, 233, 610), (2, 169, 985), (13, 34, 1325), ... There are infinitely many Markov numbers and Markov triples. Markov tree There are two simple ways to obtain a new Markov triple from an old one (''x'', ''y'', ''z''). First, one may permutation, permute the 3 numbers ''x'',''y'',''z'', so in particular one can normalize the triples so that ''x'' ≤ ''y'' ≤&nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Chiliagon
In geometry, a chiliagon () or 1,000-gon is a polygon with 1,000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation. Regular chiliagon A '' regular chiliagon'' is represented by Schläfli symbol and can be constructed as a truncated 500-gon, t, or a twice-truncated 250-gon, tt, or a thrice-truncated 125-gon, ttt. The measure of each internal angle in a regular chiliagon is 179°38'24" or \fracrad. The area of a regular chiliagon with sides of length ''a'' is given by :A = 250a^2 \cot \frac \simeq 79577.2\,a^2 This result differs from the area of its circumscribed circle by less than 4 parts per million. Because 1,000 = 23 × 53, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular chiliagon is not a constructible polygon. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
