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Pseudoprimes
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] |
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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Pseudoprime
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem. Definition Fermat's little theorem states that if p is prime and a is coprime to p, then a^-1 is divisible by p. For a positive integer a, if a composite integer x divides a^-1 then x is called a Fermat pseudoprime to base a. In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a. The false statement that all numbers that pass the Fermat primality test for base 2 are prime is called the Chinese hypothesis. The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2^ \equiv 1 (\bmod) and thus passes the Fermat primality test for the base 2. Pseudoprimes to base 2 are sometimes called Sarrus numbers, after P. F. Sarrus who discovered that 341 has this property, Poulet numbers, after P. Poulet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Somer–Lucas Pseudoprime
In mathematics, specifically number theory, an odd number, odd and composite number ''N'' is a Somer–Lucas ''d''-pseudoprime (with given ''d'' ≥ 1) if there exists a nondegenerate Lucas sequence U(P,Q) with the discriminant D=P^2-4Q, such that \gcd(N,D)=1 and the rank appearance of ''N'' in the sequence ''U''(''P'', ''Q'') is :\frac\left(N-\left(\frac\right)\right), where \left(\frac\right) is the Jacobi symbol. Applications Unlike the standard Lucas pseudoprimes, there is no known efficient primality test using the Lucas ''d''-pseudoprimes. Hence they are not generally used for computation. See also Lawrence Somer, in his 1985 thesis, also defined the Somer d-pseudoprimes. They are described in brief on page 117 of Ribenbaum 1996. References * * * * {{DEFAULTSORT:Somer-Lucas Pseudoprime Pseudoprimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Pseudoprime
In number theory, a Frobenius pseudoprime is a pseudoprime, whose definition was inspired by the quadratic Frobenius test described by Jon Grantham in a 1998 preprint and published in 2000. Frobenius pseudoprimes can be defined with respect to polynomials of degree at least 2, but they have been most extensively studied in the case of quadratic polynomials. Frobenius pseudoprimes w.r.t. quadratic polynomials The definition of Frobenius pseudoprimes with respect to a monic quadratic polynomial x^2 - Px + Q, where the discriminant D = P^2-4Q is not a square, can be expressed in terms of Lucas sequences U_n(P,Q) and V_n(P,Q) as follows. A composite number ''n'' is a Frobenius (P,Q) pseudoprime if and only if : (1) \qquad \gcd(n,2QD)=1, : (2) \qquad U_(P,Q) \equiv 0 \pmod n, and : (3) \qquad V_(P,Q) \equiv 2Q^ \pmod, where \delta=\left(\tfrac Dn\right) is the Jacobi symbol. When condition (2) is satisfied, condition (3) becomes equivalent to : (3') \qquad V_n(P,Q) \equiv P\pmod. Ther ... [...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] |
Perrin Pseudoprime
In mathematics, the Perrin numbers are a doubly infinite constant-recursive integer sequence with characteristic equation . The Perrin numbers, named after the French engineer , bear the same relationship to the Padovan sequence as the Lucas numbers do to the Fibonacci sequence. Definition The Perrin numbers are defined by the recurrence relation :\begin P(0)&=3, \\ P(1)&=0, \\ P(2)&=2, \\ P(n)&=P(n-2) +P(n-3) \mboxn>2, \end and the reverse :P(n) =P(n+3) -P(n+1) \mboxn<0. The first few terms in both directions are Perrin numbers can be expressed as sums of the three initial terms : The first fourteen prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primality Test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is '' trial division'': given an input number, n, check whether it is divisible by any prime number between 2 and \sqrt n (i.e., whether the division leaves no remainder). If so, then n is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For any divisor p \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Little Theorem
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as a^p \equiv a \pmod p. For example, if and , then , and is an integer multiple of . If is not divisible by , that is, if is coprime to , then Fermat's little theorem is equivalent to the statement that is an integer multiple of , or in symbols: a^ \equiv 1 \pmod p. For example, if and , then , and is a multiple of . Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem.. History Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy. His formulation is equivalent to the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carmichael Number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation: : b^n\equiv b\pmod for all integers . The relation may also be expressed in the form: : b^\equiv 1\pmod for all integers b that are relatively prime to . They are infinite set, 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 Prime numbers, primality. The Carmichael numbers form the subset ''K''1 of the Knödel numbers. The Carmichael numbers were named after the American mathematician Robert Daniel Carmichael, Robert Carmichael by N. G. W. H. Beeger, Nicolaas Beeger, in 1950. Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "''F'' numbers" for short. Overview Fermat's little theorem states that if p is a prime number, then for any integer , the number b^p-b is an i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Pseudoprime
In number theory, a pseudoprime is called an elliptic pseudoprime for (''E'', ''P''), where ''E'' is an elliptic curve defined over the field of rational numbers with complex multiplication by an order in \mathbb \big(\sqrt \big), having equation ''y''2 = ''x''3 + ''ax'' + ''b'' with ''a'', ''b'' integers, ''P'' being a point on ''E'' and ''n'' a natural number such that the 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 ... (−''d'' , ''n'') = −1, if . The number of elliptic pseudoprimes less than ''X'' is bounded above, for large ''X'', by : X / \exp((1/3)\log X \log\log\log X /\log\log X) \ . References * External links * Pseudoprimes {{Num-stub ... [...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. The Jacobi symbol evaluates to 0 if ''a'' and ''n'' are not coprime, so the test can alternatively be expressed as: :a^ \equiv \left(\frac\right) \neq 0 \pmod. 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''. As long as ''a'' is not a multiple of ''n'' (usually 2 ≤ ''a'' < ''n''), then if ''a'' and ''n'' are not coprime, ''n'' is definitely composite, as 1 < gcd(''a'',''n'') < ''n'' is a factor of ''n''. Properties The motivation for this definition is th ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
