|
Probable Prime
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer ''n'', choose some integer ''a'' that is not a multiple of ''n''; (typically, we choose ''a'' in the range ). Calculate . If the result is not 1, then ''n'' is composite. If the result is 1, then ''n'' is likely to be prime; ''n'' is then called a probable prime to base ''a''. A weak probable prime to base ''a'' is an integer that is a probable prime to base ''a'', but which is not a strong probable prime to base ''a'' (see below). For a fixed base ''a'', it is unusual f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miller–Rabin Primality Test
The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen primality test. It is of historical significance in the search for a polynomial-time deterministic primality test. Its probabilistic variant remains widely used in practice, as one of the simplest and fastest tests known. Gary L. Miller discovered the test in 1976; Miller's version of the test is deterministic, but its correctness relies on the unproven extended Riemann hypothesis. Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm in 1980. Mathematical concepts Similarly to the Fermat and Solovay–Strassen tests, the Miller–Rabin primality test checks whether a specific property, which is known to hold for prime values, holds for the number under testing. Strong probable primes The property is the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miller–Rabin Primality Test
The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen primality test. It is of historical significance in the search for a polynomial-time deterministic primality test. Its probabilistic variant remains widely used in practice, as one of the simplest and fastest tests known. Gary L. Miller discovered the test in 1976; Miller's version of the test is deterministic, but its correctness relies on the unproven extended Riemann hypothesis. Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm in 1980. Mathematical concepts Similarly to the Fermat and Solovay–Strassen tests, the Miller–Rabin primality test checks whether a specific property, which is known to hold for prime values, holds for the number under testing. Strong probable primes The property is the follow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baillie–PSW Primality Test
The Baillie–PSW primality test is a probabilistic primality testing algorithm that determines whether a number is composite or is a probable prime. It is named after Robert Baillie, Carl Pomerance, John Selfridge, and Samuel Wagstaff. The Baillie–PSW test is a combination of a strong Fermat probable prime test to base 2 and a strong Lucas probable prime test. The Fermat and Lucas test each have their own list of pseudoprimes, that is, composite numbers that pass the test. For example, the first ten strong pseudoprimes to base 2 are : 2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, and 52633 . The first ten strong Lucas pseudoprimes (with Lucas parameters (''P'', ''Q'') defined by Selfridge's Method A) are : 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, and 58519 . There is no known overlap between these lists of strong Fermat pseudoprimes and strong Lucas pseudoprimes, and there is even evidence that the numbers in these lists tend to be different kin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U_n(P, Q) and V_n(P, Q). More generally, Lucas sequences U_n(P, Q) and V_n(P, Q) represent sequences of polynomials in P and Q with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers . Lucas sequences are named after the French mathematician Édouard Lucas. Recurrence relations Given two integer parameters P and Q, the Lucas sequences of the first kind U_n(P,Q) and of the second kind V_n(P,Q) are defined by the recurrence relations: :\begin U_0(P,Q)&=0, \\ U_1(P,Q)&=1, \\ U_n(P,Q)&=P\cdot U_(P,Q)-Q\cdot U_(P,Q) \mbox ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Pseudoprime
Lucas pseudoprimes and Fibonacci pseudoprimes are composite integers that pass certain tests which all primes and very few composite numbers pass: in this case, criteria relative to some Lucas sequence. Baillie-Wagstaff-Lucas pseudoprimes Baillie and Wagstaff define Lucas pseudoprimes as follows: Given integers ''P'' and ''Q'', where ''P'' > 0 and D=P^2-4Q, let ''Uk''(''P'', ''Q'') and ''Vk''(''P'', ''Q'') be the corresponding Lucas sequences. Let ''n'' be a positive integer and let \left(\tfrac\right) be the Jacobi symbol. We define : \delta(n)=n-\left(\tfrac\right). If ''n'' is a prime that does not divide ''Q'', then the following congruence condition holds: If this congruence does ''not'' hold, then ''n'' is ''not'' prime. If ''n'' is ''composite'', then this congruence ''usually'' does not hold. These are the key facts that make Lucas sequences useful in primality testing. The congruence () represents one of two congruences defining a Frobenius pseudoprime. Hence, ev ... [...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 Str ... [...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] |
Solovay–Strassen Primality Test
The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen in 1977, is a probabilistic test to determine if a number is composite or probably prime. The idea behind the test was discovered by M. M. Artjuhov in 1967 (see Theorem E in the paper). This test has been largely superseded by the Baillie–PSW primality test and the Miller–Rabin primality test, but has great historical importance in showing the practical feasibility of the RSA cryptosystem. The Solovay–Strassen test is essentially an Euler–Jacobi pseudoprime test. Concepts Euler proved that for any odd prime number ''p'' and any integer ''a'', :a^ \equiv \left(\frac\right) \pmod p where \left(\tfrac\right) is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to \left(\tfrac\right), where ''n'' can be any odd integer. The Jacobi symbol can be computed in time O((log ''n'')²) using Jacobi's generalization of the law of quadratic reciprocity. G ... [...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] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. 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 natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
