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Wolstenholme Prime
In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this theorem in the 19th century. Interest in these primes first arose due to their connection with Fermat's Last Theorem. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two. The only two known Wolstenholme primes are 16843 and 2124679 . There are no other Wolstenholme primes less than 109. Definition Wolstenholme prime can be defined in a number of equivalent ways. Definition via binomial coefficients A Wolstenholme prime is a prime number ''p'' > 7 that satisfies the congruence : \equiv 1 \pmod, where t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Wolstenholme
Joseph Wolstenholme (30 September 1829 – 18 November 1891) was an English mathematician. Wolstenholme was born in Eccles near Salford, Lancashire, England, the son of a Methodist minister, Joseph Wolstenholme, and his wife, Elizabeth, ''née'' Clarke. He graduated from St John's College, Cambridge as Third Wrangler in 1850 and was elected a fellow of Christ's College in 1852. Collaborating with Percival Frost, a ''Treatise on Solid Geometry'' was published in 1863. Wolstenholme served as Examiner in 1854, 1856, and 1863 for Cambridge Mathematical Tripos, and according to Andrew Forsyth his book ''Mathematical Problems'' made a significant contribution to mathematical education: :...gathered together from many examination papers to form a volume, which was considerably amplified in later editions, they exercised a very real influence upon successive generations of undergraduates; and "Wolstenholme's Problems" have proved a help and stimulus to many students. In 1869 he resig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dots Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Zimmermann (mathematician)
Paul Zimmermann (born 13 November 1964) is a French computational mathematician, working at INRIA. Zimmermann co-authored the book Computational Mathematics with SageMath used by Mathematical students worldwide. His interests include asymptotically fast arithmetic—he wrote a book on algorithms for computer arithmetic with Richard Brent. He has developed some of the fastest available code for manipulating polynomials over GF(2), and for calculating hypergeometric constants to billions of decimal places. He is associated with the CARAMEL project to develop efficient arithmetic, in a general context and in particular in the context of algebraic curves of small genus; arithmetic on polynomials of very large degree turns out to be useful in algorithms for point-counting on such curves. He is also interested in computational number theory. In particular, he has contributed to some of the record computations in integer factorisation and discrete logarithm. He has been an acti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Arithmetica
''Acta Arithmetica'' is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sciences The Institute of Mathematics of the Polish Academy of Sciences is a research institute of the Polish Academy of Sciences.Online archives (Library of Science, Issues: 1935–2000) 1935 establishments in Poland [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Table Of Congruences
In mathematics, a congruence is an equivalence relation on the integers. The following sections list important or interesting prime-related congruences. Table of congruences characterizing special primes Other prime-related congruences There are other prime-related congruences that provide necessary and sufficient conditions on the primality of certain subsequences of the natural numbers. Many of these alternate statements characterizing primality are related to Wilson's theorem, or are restatements of this classical result given in terms of other special variants of generalized factorial functions. For instance, new variants of Wilson's theorem stated in terms of the hyperfactorials, subfactorials, and superfactorials are given in. Variants of Wilson's theorem For integers k \geq 1, we have the following form of Wilson's theorem: :(k-1)! (p-k)! \equiv (-1)^k \pmod \iff p \text If p is odd, we have that :\left(\frac\right)!^2 \equiv (-1)^ \pmod \iff p \text Clem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilson Prime
In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham. The only known Wilson primes are 5, 13, and 563 . Costa et al. Write that "the case p=5 is trivial", and credit the observation that 13 is a Wilson prime to . Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer, but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem. If any others exist, they must be greater than 2 × 1013. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval ,y/math> is about \log\log_x y. Several compute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wall–Sun–Sun Prime
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. Definition Let p be a prime number. When each term in the sequence of Fibonacci numbers F_n is reduced modulo p, the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted \pi(p). Since F_0 = 0, it follows that ''p'' divides F_. A prime ''p'' such that ''p''2 divides F_ is called a Wall–Sun–Sun prime. Equivalent definitions If \alpha(m) denotes the rank of apparition modulo m (i.e., \alpha(m) is the smallest positive index m such that m divides F_), then a Wall–Sun–Sun prime can be equivalently defined as a prime p such that p^2 divides F_. For a prime ''p'' ≠ 2, 5, the rank of apparition \alpha(p) is known to divide p - \left(\tfrac\right), where the Legendre symbol \textstyle\left(\frac\right) has the values :\left(\frac\right) = \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wieferich Prime
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians. Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the ''abc'' conjecture. , the only known Wieferich primes are 1093 and 3511 . Equivalent definitions The stronger v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (discrete)
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair dice. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a unif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Relationship
In science, an empirical relationship or phenomenological relationship is a relationship or correlation that is supported by experiment and observation but not necessarily supported by theory. Analytical solutions without a theory An empirical relationship is supported by confirmatory data irrespective of theoretical basis such as first principles. Sometimes theoretical explanations for what were initially empirical relationships are found, in which case the relationships are no longer considered empirical. An example was the Rydberg formula to predict the wavelengths of hydrogen spectral lines. Proposed in 1876, it perfectly predicted the wavelengths of the Lyman series, but lacked a theoretical basis until Niels Bohr produced his Bohr model of the atom in 1925.McMullin, Ernan (1968), “What Do Physical Models Tell Us?”, in B. van Rootselaar and J. F. Staal (eds.), Logic, Methodology and Science III. Amsterdam: North Holland, 385–396. On occasion, what was thought to be an em ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
