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90,000
90,000 (ninety thousand) is the natural number following 89,999 and preceding 90,001. It is the sum of the cubes of the first 24 positive integers, and is the square of 300. Selected numbers in the range 90,000–99,999 * 90,625 = the only five-digit automorphic number: 906252 = 8212890625 * 91,125 = 453 * 91,144 = Fine number * 92,205 = number of 23-bead necklaces (turning over is allowed) where complements are equivalent * 93,312 = Leyland number: 66 + 66. Also a 3-smooth number. * 94,249 = palindromic square: 3072 * 94,932 = Leyland number: 75 + 57 * 95,121 = Kaprekar number: 951212 = 9048004641; 90480 + 04641 = 95121 * 95,420 = number of 22-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed * 96,557 = Markov number: 52 + 64662 + 965572 = 3 × 5 × 6466 × 96557 * 97,336 = 463, the largest 5-digit cube * 98,304 = 3-smooth number * 99,066 = largest number whose square uses all of the decimal digits once: 990662 = 981407 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 factorization, 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 primality test, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
80,000
80,000 (eighty thousand) is the natural number after 79,999 and before 80,001. Selected numbers in the range 80,000–89,999 * 80,782 = Pell number ''P''14 * 81,081 = smallest abundant number ending in 1, 3, 7, or 9 * 81,181 = number of reduced trees with 25 nodes * 82,000 = the only currently known number greater than 1 that can be written in bases from 2 through 5 using only 0s and 1s. * 82,025 = number of primes \leq 2^. * 82,467 = number of square (0,1)-matrices without zero rows and with exactly 6 entries equal to 1 * 82,656 = Kaprekar number: 826562 = 6832014336; 68320 + 14336 = 82656 * 82,944 = 3-smooth number: 210 × 34 * 83,097 = Riordan number * 83,160 = the 29th highly composite number * 83,357 = Friedman prime * 83,521 = 174 * 84,187 – number of parallelogram polyominoes with 15 cells. * 84,375 = 33×55 * 84,672 = number of primitive polynomials of degree 21 over GF(2) * 85,085 = product of five consecutive primes: 5 × 7 × 11 × 13 × 17 * 85,184 = 443 * 86,400 = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Number
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base b whose square "ends" in the same digits as the number itself. Definition and properties Given a number base b, a natural number n with k digits is an automorphic number if n is a fixed point of the polynomial function f(x) = x^2 over \mathbb/b^k\mathbb, the ring of integers modulo b^k. As the inverse limit of \mathbb/b^k\mathbb is \mathbb_b, the ring of b-adic integers, automorphic numbers are used to find the numerical representations of the fixed points of f(x) = x^2 over \mathbb_b. For example, with b = 10, there are four 10-adic fixed points of f(x) = x^2, the last 10 digits of which are: : \ldots 0000000000 : \ldots 0000000001 : \ldots 8212890625 : \ldots 1787109376 Thus, the automorphic numbers in base 10 are 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 82 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Palindromic Number
A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palindromic'' is derived from palindrome, which refers to a word (such as ''rotor'' or ''racecar'') whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers (in decimal) are: : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, ... . Palindromic numbers receive most attention in the realm of recreational mathematics. A typical problem asks for numbers that possess a certain property ''and'' are palindromic. For instance: * The palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, ... . * The palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... . In any base there are infinitely many palindromic numbers, since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaprekar Number
In mathematics, a natural number in a given number base is a p-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p digits, that add up to the original number. For example, in base 10, 45 is a 2-Kaprekar number, because 45² = 2025, and 20 + 25 = 45. The numbers are named after D. R. Kaprekar. Definition and properties Let n be a natural number. Then the Kaprekar function for base b > 1 and power p > 0 F_ : \mathbb \rightarrow \mathbb is defined to be the following: :F_(n) = \alpha + \beta, where \beta = n^2 \bmod b^p and :\alpha = \frac A natural number n is a p-Kaprekar number if it is a fixed point for F_, which occurs if F_(n) = n. 0 and 1 are trivial Kaprekar numbers for all b and p, all other Kaprekar numbers are nontrivial Kaprekar numbers. The earlier example of 45 satisfies this definition with b = 10 and p = 2, because : \beta = n^2 \bmod b^p = 45^2 \bmod 10^2 = 25 : \alpha = \frac = \frac = 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 in which every prime factor is at most 7. Therefore, 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. 2-smooth numbers are simply the Power of two, powers of 2, while 5-smooth numbers are also known as regular numbers. Definition A negative and positive numbers, 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ''x''^2 ( caret) or ''x''**2 may be used in place of ''x''2. The adjective which corresponds to squaring is '' quadratic''. The square of an integer may also be called a '' square number'' or a ''perfect square''. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial is the quadratic polynomial . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that ... [...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 (''decimal fractions'') 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. That is, fractions of the form , w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Digit
A numerical digit (often shortened to just digit) or numeral is a single symbol used alone (such as "1"), or in combinations (such as "15"), to represent numbers in positional notation, such as the common base 10. The name "digit" originates from the Latin ''digiti'' meaning fingers. For any numeral system with an integer base, the number of different digits required is the absolute value of the base. For example, decimal (base 10) requires ten digits (0 to 9), and binary (base 2) requires only two digits (0 and 1). Bases greater than 10 require more than 10 digits, for instance hexadecimal (base 16) requires 16 digits (usually 0 to 9 and A to F). Overview In a basic digital system, a numeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a place value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, and summing the results. Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
