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Fermat Quotient
In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as :q_p(a) = \frac, or :\delta_p(a) = \frac. This article is about the former; for the latter see ''p''-derivation. The quotient is named after Pierre de Fermat. If the base ''a'' is coprime to the exponent ''p'' then Fermat's little theorem says that ''q''''p''(''a'') will be an integer. If the base ''a'' is also a generator of the multiplicative group of integers modulo ''p'', then ''q''''p''(''a'') will be a cyclic number, and ''p'' will be a full reptend prime. Properties From the definition, it is obvious that :\begin q_p(1) &\equiv 0 && \pmod \\ q_p(-a)&\equiv q_p(a) && \pmod\quad (\text 2 \mid p-1) \end In 1850, Gotthold Eisenstein proved that if ''a'' and ''b'' are both coprime to ''p'', then: :\begin q_p(ab) &\equiv q_p(a)+q_p(b) &&\pmod \\ q_p(a^r) &\equiv rq_p(a) &&\pmod \\ q_p(p \mp a) &\equiv q_p(a) \pm \tfrac &&\pmod \\ q_p(p \mp 1) &\equiv \pm 1 && \pmo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from 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 can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009, and is its chairman. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 370,000 sequences, and is growing by approximately 30 entries per day. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input. History Neil Sloane started coll ... [...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 stronge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ladislav Skula
Ladislav "Ladja" Skula (born June 30, 1937) is a Czech mathematician. His work spans across topology, algebraic number theory, and the theory of ordered sets. He has published over 80 papers and notable results on the Fermat quotient. He obtained his Dr.Sc. degree from Charles University in Prague with a thesis on "obor Algebra a teorie čísel" (On Algebra and Number Theory). In 1991, he was appointed professor at the Masaryk University in Brno, where he is now emeritus professor ''Emeritus/Emerita'' () is an honorary title granted to someone who retires from a position of distinction, most commonly an academic faculty position, but is allowed to continue using the previous title, as in "professor emeritus". In some c .... Selected publications * * * * * * External links *Skula'homepageat Masaryk University {{DEFAULTSORT:Skula, Ladislav Czech mathematicians Number theorists Living people 1937 births Charles University alumni Academic staff of Masaryk Univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Whitbread Lee Glaisher
James Whitbread Lee Glaisher (5 November 1848, in Lewisham — 7 December 1928, in Cambridge) was a prominent English mathematician and astronomer. He is known for Glaisher's theorem, an important result in the field of integer partitions, and for the Glaisher–Kinkelin constant, a number important in both mathematics and physics. He was a passionate collector of English ceramics and valentines, much of which he bequeathed to the Fitzwilliam Museum in Cambridge. Life He was born in Lewisham in Kent on 5 November 1848 the son of the eminent astronomer James Glaisher and his wife, Cecilia Louisa Belville. His mother was a noted photographer. He was educated at St Paul's School from 1858. He became somewhat of a school celebrity in 1861 when he made two hot-air balloon ascents with his father to study the stratosphere. He won a Campden Exhibition Scholarship allowing him to study at Trinity College, Cambridge, where he was second wrangler in 1871 and was made a Fe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Multiplicative Inverse
In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus .. In the standard notation of modular arithmetic this congruence is written as :ax \equiv 1 \pmod, which is the shorthand way of writing the statement that divides (evenly) the quantity , or, put another way, the remainder after dividing by the integer is 1. If does have an inverse modulo , then there is an infinite number of solutions of this congruence, which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to (i.e., in 's congruence class) has any element of 's congruence class as a modular multiplicative inverse. Using the notation of \overline to indicate the congruence class containing , this can be expressed by saying that the ''modulo multiplicative inverse'' of the congruence class \overline is the congruence class \overline such that: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilson Quotient
The Wilson quotient ''W''(''p'') is defined as: :W(p) = \frac If ''p'' is a prime number, the quotient is an integer by Wilson's theorem; moreover, if ''p'' is composite, the quotient is not an integer. If ''p'' divides ''W''(''p''), it is called a Wilson prime. The integer values of ''W''(''p'') are : : ''W''(2) = 1 : ''W''(3) = 1 : ''W''(5) = 5 : ''W''(7) = 103 : ''W''(11) = 329891 : ''W''(13) = 36846277 : ''W''(17) = 1230752346353 : ''W''(19) = 336967037143579 : ... It is known that :W(p)\equiv B_-B_\pmod, :p-1+ptW(p)\equiv pB_\pmod{p^2}, where B_k is the ''k''-th Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent function .... Note that the first relation comes from the second one by s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathias Lerch
Mathias Lerch or Matyáš Lerch (; 20 February 1860, Milínov – 3 August 1922, Sušice) was a Czech mathematician who published about 250 papers, largely on mathematical analysis and number theory. He studied in Prague (Czech Technical University) and Berlin; subsequently held teaching positions at the University of Fribourg in Switzerland, the Brno University of Technology in Brno, and finally at then newly founded (1920) Masaryk University in Brno where he became its first mathematics professor. In 1900, he was awarded the Grand Prize of the French Academy of Sciences for his number-theoretic work. The Lerch zeta function is named after him, as is the Appell–Lerch sum. His doctoral students include Michel Plancherel and Otakar Borůvka Otakar Borůvka (10 May 1899 – 22 July 1995) was a Czech mathematician. He is best known for his work in graph theory.. Education and career Borůvka was born in Uherský Ostroh, a town in Moravia, Austria-Hungary (today in the Cze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Bachmann
Paul Gustav Heinrich Bachmann (22 June 1837 – 31 March 1920) was a German mathematician. Life Bachmann studied mathematics at the university of his native city of Berlin and received his doctorate in 1862 for his thesis on group theory. He then went to Breslau to study for his habilitation, which he received in 1864 for his thesis on Complex Units. He was a professor at Breslau and later at Münster. Works *''Zahlentheorie'', Bachmann's work on number theory in five volumes (1872-1923): **Vol. I: Die Elemente der Zahlentheorie' (1892) **Vol. II: Analytische Zahlentheorie' (1894), a work on analytic number theory in which Big O notation was first introduced **Vol. III: Die Lehre von der Kreistheilung und ihre Beziehungen zur Zahlentheorie' (first published in 1872) **Vol. IV (Part 1): Die Arithmetik der quadratischen Formen' (1898) **Vol. IV (Part 2): Die Arithmetik der quadratischen Formen' (posthumously published in 1923) **Vol. V: Allgemeine Arithmetik der Zahlenkörper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another proposition; it might also be used more casually to refer to something which naturally or incidentally accompanies something else. Overview In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term ''corollary'', rather than ''proposition'' or ''theorem'', is intrinsically subjective. More formally, proposition ''B'' is a corollary of proposition ''A'', if ''B'' can be readily deduced from ''A'' or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem easier to use and apply, even though its importance is generally considered to be secondary to that of the theorem. In particular, ''B'' is unlikely to be te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dmitry Mirimanoff
Dmitry Semionovitch Mirimanoff (; 13 September 1861, Pereslavl-Zalessky, Russia – 5 January 1945, Geneva, Switzerland) was a member of the Moscow Mathematical Society in 1897. And later became a doctor of mathematical sciences in 1900, in Geneva, and taught at the universities of Geneva and Lausanne. Mirimanoff made notable contributions to axiomatic set theory and to number theory (relating specifically to Fermat's Last Theorem, on which he corresponded with Albert Einstein before the First World WarJean A. Mirimanoff. Private correspondence with Anton Lokhmotov. (2009)). In 1917, he introduced, though not as explicitly as John von Neumann later, the cumulative hierarchy of sets and the notion of von Neumann ordinals; although he introduced a notion of regular (and well-founded set) he did not consider regularity as an axiom, but also explored what is now called non-well-founded set theory and had an emergent idea of what is now called bisimulation. Life Dmitry Semion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
