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91 (number)
91 (ninety-one) is the natural number following 90 (number), 90 and preceding 92 (number), 92. In mathematics 91 is: * the twenty-seventh distinct semiprime and the second of the form (7.q), where q is a higher prime. * the aliquot sum of 91 is 21; itself a semiprime, within an aliquot sequence of two composite numbers (91, 21, 11, 1, 0) to the prime in the 11-aliquot tree. 91 is the fourth composite number in the 11-aliquot tree. (91, 51, 21, 18). * the 13th triangular number. * a hexagonal number, one of the few such numbers to also be a centered hexagonal number. * a centered nonagonal number. * a centered cube number. * a square pyramidal number, being the sum of the squares of the first six integers. * the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): . (See 1729 (number), 1729 for more details). This implies that 91 is the second cabtaxi ... [...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] |
Centered Cube Number
A centered cube number is a centered figurate number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with points on the square faces of the th layer. Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has points along each of its edges. The first few centered cube numbers are : 1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, ... . Formulas The centered cube number for a pattern with concentric layers around the central point is given by the formula :n^3 + (n + 1)^3 = (2n+1)\left(n^2+n+1\right). The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as :\binom-\binom = (n^2+1)+(n^2+2)+\cdots+(n+1)^2. Properties Because of the factorization , it is impossible for a centered cube number to be a prime number. The only centered cube numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McCarthy 91 Function
The McCarthy 91 function is a recursive function, defined by the computer scientist John McCarthy as a test case for formal verification within computer science. The McCarthy 91 function is defined as :M(n)=\begin n - 10, & \mboxn > 100\mbox \\ M(M(n+11)), & \mboxn \le 100\mbox \end The results of evaluating the function are given by ''M''(''n'') = 91 for all integer arguments ''n'' ≤ 100, and ''M''(''n'') = ''n'' − 10 for ''n'' > 100. Indeed, the result of M(101) is also 91 (101 - 10 = 91). All results of M(n) after n = 101 are continually increasing by 1, e.g. M(102) = 92, M(103) = 93. History The 91 function was introduced in papers published by Zohar Manna, Amir Pnueli and John McCarthy in 1970. These papers represented early developments towards the application of formal methods to program verification. The 91 function was chosen for being nested-recursive (contrasted with single recursion, such as defining f(n) by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius Colleg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime
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] |
Repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary). Any such number can be represented as follows \underbrace_ = \frac Where nn is the concatenation of n with n. k the number of concatenated n. nn can be represented mathematically as n\cdot\left(10^+1\right) for n = 23 and k = 5, the formula will look like this \frac = \frac = \underbrace_ However, 2323232323 is not a repdigit. Also, any number can be decomposed into the sum and difference of the repdigit numbers. For example 3453455634 = 3333333333 + (111111111 + (99999 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoprime
A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy. Some sources use the term pseudoprime to describe all probable primes, both composite numbers and actual primes. Pseudoprimes are of primary importance in public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Carl Pomerance estimated in 1988 that it would cost $10 million to factor a number with 144 digits, and $100 billion to factor a 200-digit number (the cost today is dramatically lower but still prohibitively high). But finding two large prime numbers as needed for this use is also expensive, so various probabilistic primality tests are used, some of which in rare cases inappropriately deliver composite numbers instead of primes. On the other hand, deterministic primality tests, such as the AKS primality test, do not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cabtaxi Number
In number theory, the -th cabtaxi number, typically denoted , is defined as the smallest positive integer that can be written as the sum of two ''positive or negative or 0'' cubes in ways. Such numbers exist for all , which follows from the analogous result for taxicab numbers. Known cabtaxi numbers Only 10 cabtaxi numbers are known : \begin \mathrm(1) =& \ 1 \\ &= 1^3 + 0^3 \\ pt \mathrm(2) =& \ 91 \\ &= 3^3 + 4^3 \\ &= 6^3 - 5^3 \\ pt \mathrm(3) =& \ 728 \\ &= 6^3 + 8^3 \\ &= 9^3 - 1^3 \\ &= 12^3 - 10^3 \\ pt \mathrm(4) =& \ 2741256 \\ &= 108^3 + 114^3 \\ &= 140^3 - 14^3 \\ &= 168^3 - 126^3 \\ &= 207^3 - 183^3 \\ pt \mathrm(5) =& \ 6017193 \\ &= 166^3 + 113^3 \\ &= 180^3 + 57^3 \\ &= 185^3 - 68^3 \\ &= 209^3 - 146^3 \\ &= 246^3 - 207^3 \\ pt \mathrm(6) =& \ 1412774811 \\ &= 963^3 + 804^3 \\ &= 1134^3 - 357^3 \\ &= 1155^3 - 504^3 \\ &= 1246^3 - 805^3 \\ &= 2115^3 - 2004^3 \\ &= 4746^3 - 4725^3 \\ pt \mathrm(7 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1729 (number)
1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic positive integers in two different ways. It is known as the Ramanujan number or Hardy–Ramanujan number after G. H. Hardy and Srinivasa Ramanujan. As a natural number 1729 is composite, the squarefree product of three prime numbers 7 × 13 × 19. It has as factors 1, 7, 13, 19, 91, 133, 247, and 1729. It is the third Carmichael number, and the first Chernick–Carmichael number. Furthermore, it is the first in the family of absolute Euler pseudoprimes, a subset of Carmichael numbers. 1729 is divisible by 19, the sum of its digits, making it a harshad number in base 10. 1729 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based. This is an example of a galactic algorithm. 1729 can be expressed as the quadratic form. Investigating pairs of its distinct integer-valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Two Cubes
In mathematics, the sum of two cubes is a cubed number added to another cubed number. Factorization Every sum of cubes may be factored according to the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2) in elementary algebra. Binomial numbers generalize this factorization to higher odd powers. Proof Starting with the expression, a^2-ab+b^2 and multiplying by (a+b)(a^2-ab+b^2) = a(a^2-ab+b^2) + b(a^2-ab+b^2). distributing ''a'' and ''b'' over a^2-ab+b^2, a^3 - a^2 b + ab^2 + a^2b - ab^2 + b^3 and canceling the like terms, a^3 + b^3. Similarly for the difference of cubes, \begin (a-b)(a^2+ab+b^2) & = a(a^2+ab+b^2) - b(a^2+ab+b^2) \\ & = a^3 + a^2 b + ab^2 \; - a^2b - ab^2 - b^3 \\ & = a^3 - b^3. \end "SOAP" mnemonic The mnemonic "SOAP", short for "Same, Opposite, Always Positive", helps recall of the signs: : Fermat's last theorem Fermat's last theorem in the case of exponent 3 states that the sum of two non-zero integer cubes does not result in a non-zero integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Pyramidal Number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid (geometry), pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes. As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first n positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number. History The pyramidal numbers were one of the few types of three-dimensional fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
