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Euler Pseudoprime
In arithmetic, an odd composite integer ''n'' is called an Euler pseudoprime to base ''a'', if ''a'' and ''n'' are coprime, and : a^ \equiv \pm 1\pmod (where ''mod'' refers to the modulo operation). The motivation for this definition is the fact that all prime numbers ''p'' satisfy the above equation which can be deduced from Fermat's little theorem. Fermat's theorem asserts that if ''p'' is prime, and coprime to ''a'', then ''a''''p''−1 ≡ 1 (mod ''p''). Suppose that ''p''>2 is prime, then ''p'' can be expressed as 2''q'' + 1 where ''q'' is an integer. Thus, ''a''(2''q''+1) − 1 ≡ 1 (mod ''p''), which means that ''a''2''q'' − 1 ≡ 0 (mod ''p''). This can be factored as (''a''''q'' − 1)(''a''''q'' + 1) ≡ 0 (mod ''p''), which is equivalent to ''a''(''p''−1)/2 ≡ ±1 (mod ''p''). The equation can be tested rather quickly, which can be used for probabilistic primality testing. These tests are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms, which are highly important to the field of mathematical logic today. History The prehistory of arithmetic is limited to a small number of artifacts, which may indicate the conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations: addition, subtraction, multiplication, and division, as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra (structure), Boolean algebra under the subset relation, in which the join and meet are given by Intersection (set theory), intersection and Union (set theory), union, and the subset relation itself is the Inclusion (Boolean algebra), Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys. Modular exponentiation is the remainder when an integer (the base) is raised to the power (the exponent), and divided by a positive integer (the modulus); that is, . From the definition of division, it follows that . For example, given , and , dividing by leaves a remainder of . Modular exponentiation can be performed with a ''negative'' exponent by finding the modular multiplicative inverse of modulo using the extended Euclidean algorithm. That is: :, where and . Modular exponentiation is efficient to compute, even for very large integers. On the other hand, computing the modular discrete logarithm – that is, finding the exponent when given , , and – is believed to be difficult. This one-way function behavior makes modular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lua (programming Language)
Lua ( ; from meaning ''moon'') is a lightweight, high-level, multi-paradigm programming language designed primarily for embedded use in applications. Lua is cross-platform, since the interpreter of compiled bytecode is written in ANSI C, and Lua has a relatively simple C API to embed it into applications. Lua originated in 1993 as a language for extending software applications to meet the increasing demand for customization at the time. It provided the basic facilities of most procedural programming languages, but more complicated or domain-specific features were not included; rather, it included mechanisms for extending the language, allowing programmers to implement such features. As Lua was intended to be a general embeddable extension language, the designers of Lua focused on improving its speed, portability, extensibility, and ease-of-use in development. History Lua was created in 1993 by Roberto Ierusalimschy, Luiz Henrique de Figueiredo, and Waldemar Celes, membe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Pseudoprime
A strong pseudoprime is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making them "pseudoprimes". Unlike the Fermat pseudoprimes, for which there exist numbers that are pseudoprimes to all coprime bases (the Carmichael numbers), there are no composites that are strong pseudoprimes to all bases. Motivation and first examples Let us say we want to investigate if ''n'' = 31697 is a probable prime (PRP). We pick base ''a'' = 3 and, inspired by Fermat's little theorem, calculate: : 3^ \equiv 1 \pmod This shows 31697 is a Fermat PRP (base 3), so we may suspect it is a prime. We now repeatedly halve the exponent: : 3^ \equiv 1 \pmod : 3^ \equiv 1 \pmod : 3^ \equiv 28419 \pmod The first couple of times do not yield anything interesting (the result was still 1 modulo 31697), but at exponent 3962 we see a result that is neither 1 nor minus 1 (i.e. 31696) modulo 31697. This proves 31697 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Jacobi Pseudoprime
In number theory, an odd integer ''n'' is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base ''a'', if ''a'' and ''n'' are coprime, and :a^ \equiv \left(\frac\right)\pmod where \left(\frac\right) is the Jacobi symbol. If ''n'' is an odd composite integer that satisfies the above congruence, then ''n'' is called an Euler–Jacobi pseudoprime (or, more commonly, an Euler pseudoprime) to base ''a''. Properties The motivation for this definition is the fact that all prime numbers ''n'' satisfy the above equation, as explained in the Euler's criterion article. The equation can be tested rather quickly, which can be used for probabilistic primality testing. These tests are over twice as strong as tests based on Fermat's little theorem. Every Euler–Jacobi pseudoprime is also a Fermat pseudoprime and an Euler pseudoprime. There are no numbers which are Euler–Jacobi pseudoprimes to all bases as Carmichael numbers are. Solovay and Stra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Symbol
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a Jacobi symbol of −1 is a quadratic residue, and if ''k'' is a quadratic residue modulo a coprime ''n'', then , but not all entries with a Jacobi symbol of 1 (see the and rows) are quadratic residues. Notice also that when either ''n'' or ''k'' is a square, all values are nonnegative. The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are important in cryptography. Definition For any integer ''a'' and any positive odd integer ''n'', the Jacobi symbol is defined as the product of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below). Overview Definition The ''greatest common divisor'' (GCD) of two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the largest s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1729 (number)
1729 is the natural number following 1728 and preceding 1730. It is a taxicab number, and is variously known as Ramanujan's number and the Ramanujan-Hardy number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation: The two different ways are: : 1729 = 13 + 123 = 93 + 103 The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729; 1991 = 1729). :91 = 63 + (−5)3 = 43 + 33 Numbers that are the smallest number that can be expressed as the sum of two cubes in ''n'' distinct ways have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657. A commemorative plaque now appears at the site of the Ramanujan-Hardy inciden ... [...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] |
Number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called ''numerals''; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a ''numeral'' is not clearly distinguished from the ''number'' th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
