|
Prime95 28
Prime95, also distributed as the command-line utility mprime for FreeBSD and Linux, is a freeware application written by George Woltman. It is the official client of the Great Internet Mersenne Prime Search (GIMPS), a volunteer computing project dedicated to searching for Mersenne primes. It is also used in overclocking to test for system stability. Although most of its source code is available, Prime95 is not free and open-source software because its end-user license agreement states that if the software is used to find a prime qualifying for a bounty offered by the Electronic Frontier Foundation, then that bounty will be claimed and distributed by GIMPS. Finding Mersenne primes by volunteer computing Prime95 tests numbers for primality using the Fermat primality test (referred to internally as PRP, or "probable prime"). For much of its history, it used the Lucas–Lehmer primality test, but the availability of Lucas–Lehmer assignments was deprecated in April 2021 to increase ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Windows 7
Windows 7 is a major release of the Windows NT operating system developed by Microsoft. It was released to manufacturing on July 22, 2009, and became generally available on October 22, 2009. It is the successor to Windows Vista, released nearly three years earlier. It remained an operating system for use on personal computers, including home and business desktops, laptops, tablet PCs and media center PCs, and itself was replaced in November 2012 by Windows 8, the name spanning more than three years of the product. Until April 9, 2013, Windows 7 original release included updates and technical support, after which installation of Service Pack 1 was required for users to receive support and updates. Windows 7's server counterpart, Windows Server 2008 R2, was released at the same time. The last supported version of Windows based on this operating system was released on July 1, 2011, entitled Windows Embedded POSReady 7. Extended support ended on January 14, 2020, over ten years a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
End-user License Agreement
An end-user license agreement or EULA () is a legal contract between a software supplier and a customer or end-user, generally made available to the customer via a retailer acting as an intermediary. A EULA specifies in detail the rights and restrictions which apply to the use of the software. Form contracts for digital services (such as terms of service and privacy policies) were traditionally presented on paper (see shrink-wrap agreement) but are now often presented digitally via browsewrap or clickwrap formats. As the user may not see the agreement until after they have already purchased or engaged with the software, these documents may be contracts of adhesion. Software companies often make special agreements with large businesses and government entitles that include support contracts and specially drafted warranties. Many EULAs assert extensive liability limitations. Most commonly, an EULA will attempt to hold harmless the software licensor in the event that the software cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime95 28
Prime95, also distributed as the command-line utility mprime for FreeBSD and Linux, is a freeware application written by George Woltman. It is the official client of the Great Internet Mersenne Prime Search (GIMPS), a volunteer computing project dedicated to searching for Mersenne primes. It is also used in overclocking to test for system stability. Although most of its source code is available, Prime95 is not free and open-source software because its end-user license agreement states that if the software is used to find a prime qualifying for a bounty offered by the Electronic Frontier Foundation, then that bounty will be claimed and distributed by GIMPS. Finding Mersenne primes by volunteer computing Prime95 tests numbers for primality using the Fermat primality test (referred to internally as PRP, or "probable prime"). For much of its history, it used the Lucas–Lehmer primality test, but the availability of Lucas–Lehmer assignments was deprecated in April 2021 to increase ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Largest Known Prime Number
The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. A prime number is a positive integer, excluding 1, with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime. Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilise a specialised primality test that is faster than the general one. , the eight largest known primes are Mersenne primes. The last seventeen record primes were Mersenne primes. The binary representation of any Mersenne prime is composed of all 1's, since the binary form of 2''k'' − 1 is simply ''k'' 1's. Current record The record is currently held by with 24,862,048 digits, found by GIMPS in December 2018. The first and last 120 digits of its val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphics Processing Unit
A graphics processing unit (GPU) is a specialized electronic circuit designed to manipulate and alter memory to accelerate the creation of images in a frame buffer intended for output to a display device. GPUs are used in embedded systems, mobile phones, personal computers, workstations, and game consoles. Modern GPUs are efficient at manipulating computer graphics and image processing. Their parallel structure makes them more efficient than general-purpose central processing units (CPUs) for algorithms that process large blocks of data in parallel. In a personal computer, a GPU can be present on a video card or embedded on the motherboard. In some CPUs, they are embedded on the CPU die. In the 1970s, the term "GPU" originally stood for ''graphics processor unit'' and described a programmable processing unit independently working from the CPU and responsible for graphics manipulation and output. Later, in 1994, Sony used the term (now standing for ''graphics processing unit'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double-precision Floating-point Format
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width), even if at the cost of precision. Double precision may be chosen when the range or precision of single precision would be insufficient. In the IEEE 754-2008 standard, the 64-bit base-2 format is officially referred to as binary64; it was called double in IEEE 754-1985. IEEE 754 specifies additional floating-point formats, including 32-bit base-2 ''single precision'' and, more recently, base-10 representations. One of the first programming languages to provide single- and double-precision floating-point data types was Fortran. Before the widespread adoption of IEEE 754-1985, the representation and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Single-precision Floating-point Format
Single-precision floating-point format (sometimes called FP32 or float32) is a computer number format, usually occupying 32 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 231 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2−23) × 2127 ≈ 3.4028235 × 1038. All integers with 7 or fewer decimal digits, and any 2''n'' for a whole number −149 ≤ ''n'' ≤ 127, can be converted exactly into an IEEE 754 single-precision floating-point value. In the IEEE 754-2008 standard, the 32-bit base-2 format is officially referred to as binary32; it was called single in IEEE 754-1985. IEEE 754 specifies additional floating-point types, such as 64-bit base-2 ''double prec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trial Division
Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer ''n'', the integer to be factored, can be divided by each number in turn that is less than ''n''. For example, for the integer , the only numbers that divide it are 1, 2, 3, 4, 6, 12. Selecting only the largest powers of primes in this list gives that . Trial division was first described by Fibonacci in his book ''Liber Abaci'' (1202). Method Given an integer ''n'' (''n'' refers to "the integer to be factored"), the trial division consists of systematically testing whether ''n'' is divisible by any smaller number. Clearly, it is only worthwhile to test candidate factors less than ''n'', and in order from two upwards because an arbitrary ''n'' is more likely to be divisible by two than by three, and so on. With this ordering, there is no point in testing for divisibility by four if the number has already b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization
In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind. For example, is a factorization of the integer , and is a factorization of the polynomial . Factorization is not usually considered meaningful within number systems possessing division ring, division, such as the real number, real or complex numbers, since any x can be trivially written as (xy)\times(1/y) whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by Greek mathematics, ancient Greek mathematicians in the case of integers. They proved the fundamental theorem o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Williams's P + 1 Algorithm
In computational number theory, Williams's ''p'' + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms. It was invented by Hugh C. Williams in 1982. It works well if the number ''N'' to be factored contains one or more prime factors ''p'' such that ''p'' + 1 is smooth, i.e. ''p'' + 1 contains only small factors. It uses Lucas sequences to perform exponentiation in a quadratic field. It is analogous to Pollard's ''p'' − 1 algorithm. Algorithm Choose some integer ''A'' greater than 2 which characterizes the Lucas sequence: :V_0=2, V_1=A, V_j=AV_-V_ where all operations are performed modulo ''N''. Then any odd prime ''p'' divides \gcd(N,V_M-2) whenever ''M'' is a multiple of p-(D/p), where D=A^2-4 and (D/p) is the Jacobi symbol. We require that (D/p)=-1, that is, ''D'' should be a quadratic non-residue modulo ''p''. But as we don't know ''p'' beforehand, more than one value of ''A'' may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lenstra Elliptic-curve Factorization
The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra. Practically speaking, ECM is considered a special-purpose factoring algorithm, as it is most suitable for finding small factors. , it is still the best algorithm for divisors not exceeding 50 to 60 digits, as its running time is dominated by the size of the smallest factor ''p'' rather than by the size of the number ''n'' to be factored. Frequently, ECM is used to remove small factors from a very large integer with many factors; if the remaining integer is still composite, then it has only large factors and is factored ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollard's P − 1 Algorithm
Pollard's ''p'' − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm. The factors it finds are ones for which the number preceding the factor, ''p'' − 1, is powersmooth; the essential observation is that, by working in the multiplicative group modulo a composite number ''N'', we are also working in the multiplicative groups modulo all of ''Ns factors. The existence of this algorithm leads to the concept of safe primes, being primes for which ''p'' − 1 is two times a Sophie Germain prime ''q'' and thus minimally smooth. These primes are sometimes construed as "safe for cryptographic purposes", but they might be ''unsafe'' — in current recommendations for cryptographic strong primes (''e.g.'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
