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Primes In Arithmetic Progression
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a_n = 3 + 4n for 0 \le n \le 2. According to the Green–Tao theorem, there exist arbitrarily long sequences of primes in arithmetic progression. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form an + b, where ''a'' and ''b'' are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For integer ''k'' ≥ 3, an AP-''k'' (also called PAP-''k'') is any sequence of ''k'' primes in arithmetic progression. An AP-''k'' can be written as ''k'' primes of the form ''a''·''n'' + ''b'', for fixed integers ''a'' (called th ... [...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] |
Sexy Prime
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and . The term "sexy prime" is a pun stemming from the Latin word for six: . If or (where is the lower prime) is also prime, then the sexy prime is part of a prime triplet. In August 2014 the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin or sexy primes. Primorial ''n''# notation As used in this article, # stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ . Types of groupings Sexy prime pairs The sexy primes (sequences and in OEIS) below 500 are: :(5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributed Computing
A distributed system is a system whose components are located on different computer network, networked computers, which communicate and coordinate their actions by message passing, passing messages to one another from any system. Distributed computing is a field of computer science that studies distributed systems. The components of a distributed system interact with one another in order to achieve a common goal. Three significant challenges of distributed systems are: maintaining concurrency of components, overcoming the clock synchronization, lack of a global clock, and managing the independent failure of components. When a component of one system fails, the entire system does not fail. Examples of distributed systems vary from service-oriented architecture, SOA-based systems to massively multiplayer online games to peer-to-peer, peer-to-peer applications. A computer program that runs within a distributed system is called a distributed program, and ''distributed programming' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Prime
In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, given a prime number p_n, where is its index in the ordered set of prime numbers, :p_n = . For example, 53 is the sixteenth prime; the fifteenth and seventeenth primes, 47 and 59, add up to 106, and half of that is 53; thus 53 is a balanced prime. Examples The first few balanced primes are 5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903 . Infinitude It is conjectured that there are infinitely many balanced primes. Three consecutive primes in arithmetic progression is sometimes called a CPAP-3. A balanced prime is by definition the second prime in a CPAP-3. the largest known CPAP-3 has 10546 digits and was found by David Broadhurst. It is: [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''. A ''decimal numeral'' (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in or ). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in " is the approximation of to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value. The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form , where is an integer, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duron
Duron is a line of budget x86-compatible microprocessors manufactured by AMD. Released on June 19, 2000 as a lower-cost offering to complement AMD's then mainstream performance Athlon processor line, it also competed with rival chipmaker Intel's Pentium III and Celeron processor offerings. The Duron brand name was retired in 2004, succeeded by the Sempron line of processors as AMD's budget offering. Performance The original Duron processors were derived from AMD's mainstream ''Athlon'' Thunderbird processors, the primary difference being a reduction in L2 cache size to 64 KB from the Athlon's 256 KB. This was a relatively severe reduction, making it even smaller than the 128 KB L2 available on Intel's competing budget Celeron line. However, the originating Thunderbird architecture already featured one of the largest L1 caches at 128 KB (which was not reduced in the Duron) and also introduced AMD's switch to an exclusive cache design which effectively unifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentium D
Pentium D is a range of desktop 64-bit x86-64 processors based on the NetBurst microarchitecture, which is the dual-core variant of the Pentium 4 manufactured by Intel. Each CPU comprised two dies, each containing a single core, residing next to each other on a multi-chip module package. The brand's first processor, codenamed ''Smithfield'' and manufactured on the 90 nm process, was released on May 25, 2005, followed by the 65 nm ''Presler'' nine months later. By 2004, the NetBurst processors reached a clock speed barrier at 3.8 GHz due to a thermal (and power) limit exemplified by the ''Presler's'' 130 watt thermal design power (a higher TDP requires additional cooling that can be prohibitively noisy or expensive). The future belonged to more energy efficient and slower clocked dual-core CPUs on a single die instead of two. However, the Pentium D did not offer significant upgrades in design, still resulting in relatively high power consumption. The final shipment date of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Athlon
Athlon is the brand name applied to a series of x86-compatible microprocessors designed and manufactured by Advanced Micro Devices (AMD). The original Athlon (now called Athlon Classic) was the first seventh-generation x86 processor and the first desktop processor to reach speeds of one gigahertz (GHz). It made its debut as AMD's high-end processor brand on June 23, 1999. Over the years AMD has used the Athlon name with the 64-bit Athlon 64 architecture, the Athlon II, and Accelerated Processing Unit (APU) chips targeting the Socket AM1 desktop System on a chip, SoC architecture, and Socket AM4 Zen microarchitecture. The modern Zen-based Athlon with a Radeon, Radeon Graphics processor was introduced in 2019 as AMD's highest-performance entry-level processor. Athlon comes from the Ancient Greek (''athlon''), meaning "(sport) contest", or "prize of a contest", or "place of a contest; arena". With the Athlon name originally used for AMD's high-end processors, AMD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Athlon 64
The Athlon 64 is a ninth-generation, AMD64-architecture microprocessor produced by Advanced Micro Devices (AMD), released on September 23, 2003. It is the third processor to bear the name ''Athlon'', and the immediate successor to the Athlon XP. The second processor (after the Opteron) to implement the AMD64 architecture and the first 64-bit processor targeted at the average consumer, it was AMD's primary consumer CPU, and primarily competed with Intel's Pentium 4, especially the ''Prescott'' and ''Cedar Mill'' core revisions. It is AMD's first K8, eighth-generation processor core for desktop and mobile computers. Despite being natively 64-bit, the AMD64 architecture is backward-compatible with 32-bit x86 instructions. Athlon 64s have been produced for Socket 754, Socket 939, Socket 940, and Socket AM2. The line was succeeded by the dual-core Athlon 64 X2 and Athlon X2 lines. Background The Athlon 64 was originally codenamed ''ClawHammer'' by AMD, and was referred to as such in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell Microprocessor
Cell is a multi-core microprocessor microarchitecture that combines a general-purpose PowerPC core of modest performance with streamlined coprocessing elements which greatly accelerate multimedia and vector processing applications, as well as many other forms of dedicated computation. It was developed by Sony, Toshiba, and IBM, an alliance known as "STI". The architectural design and first implementation were carried out at the STI Design Center in Austin, Texas over a four-year period beginning March 2001 on a budget reported by Sony as approaching US$400 million. Cell is shorthand for Cell Broadband Engine Architecture, commonly abbreviated ''CBEA'' in full or ''Cell BE'' in part. The first major commercial application of Cell was in Sony's PlayStation 3 game console, released in 2006. In May 2008, the Cell-based IBM Roadrunner supercomputer became the first TOP500 LINPACK sustained 1.0 petaflops system. Mercury Computer Systems also developed designs based on the Cell. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CUDA
CUDA (or Compute Unified Device Architecture) is a parallel computing platform and application programming interface (API) that allows software to use certain types of graphics processing units (GPUs) for general purpose processing, an approach called general-purpose computing on GPUs (GPGPU). CUDA is a software layer that gives direct access to the GPU's virtual instruction set and parallel computational elements, for the execution of compute kernels. CUDA is designed to work with programming languages such as C, C++, and Fortran. This accessibility makes it easier for specialists in parallel programming to use GPU resources, in contrast to prior APIs like Direct3D and OpenGL, which required advanced skills in graphics programming. CUDA-powered GPUs also support programming frameworks such as OpenMP, OpenACC and OpenCL; and HIP by compiling such code to CUDA. CUDA was created by Nvidia. When it was first introduced, the name was an acronym for Compute Unified Device Architectur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nvidia
Nvidia CorporationOfficially written as NVIDIA and stylized in its logo as VIDIA with the lowercase "n" the same height as the uppercase "VIDIA"; formerly stylized as VIDIA with a large italicized lowercase "n" on products from the mid 1990s to early-mid 2000s. Though unofficial, second letter capitalization of NVIDIA, i.e. nVidia, may be found within enthusiast communities and publications. ( ) is an American multinational technology company incorporated in Delaware and based in Santa Clara, California. It is a software and fabless company which designs graphics processing units (GPUs), application programming interface (APIs) for data science and high-performance computing as well as system on a chip units (SoCs) for the mobile computing and automotive market. Nvidia is a global leader in artificial intelligence hardware and software. Its professional line of GPUs are used in workstations for applications in such fields as architecture, engineering and construction, media ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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