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Topic: Modular arithmetic theory

Related Topics

Modular arithmetic

Euler pseudoprime

Carmichael number

Fibonacci pseudoprime

Ring (mathematics)

Prime number

Chinese remainder theorem

Coprime

RSA

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	modular-arithmetic
		(def-theorem mod-n-range-in-zz_mod "forall(a:zz, #(a mod modulus,zz_mod))" (theory arithmetic-mod-n) (proof ((apply-macete-with-minor-premises zz_mod-defining-axiom_arithmetic-mod-n) beta-reduce-with-minor-premises (move-to-sibling 1) (apply-macete-with-minor-premises mod-of-integer-is-integer) simplify direct-inference (cut-with-single-formula "forsome(k:zz, a mod modulus =k)") (move-to-sibling 1) (instantiate-existential ("a mod modulus")) simplify (cut-with-single-formula "#(a mod modulus,zz)") (apply-macete-with-minor-premises mod-of-integer-is-integer) simplify (antecedent-inference "with(p:prop,p);") (backchain "with(p:prop,p);") (backchain "with(p:prop,p);") (incorporate-antecedent "with(p:prop,p);") (apply-macete-with-minor-premises mod-characterization) direct-and-antecedent-inference-strategy)))
		(def-theorem _mod-characterization "forall(a,b,c:zz_mod, _mod(a,b)=c iff (0<=c and ctheory arithmetic-mod-n) (proof ((unfold-single-defined-constant-globally *_mod) (apply-macete-with-minor-premises congruence-characterization) direct-and-antecedent-inference-strategy (incorporate-antecedent "with(p:prop,p);") (apply-macete-with-minor-premises mod-characterization) direct-and-antecedent-inference-strategy (incorporate-antecedent "with(c:zz_mod,r:rr,r=c);") (apply-macete-with-minor-premises mod-characterization) direct-and-antecedent-inference-strategy (cut-with-single-formula "c mod modulus = c") (apply-macete-with-minor-premises mod-characterization) (unfold-single-defined-constant (0) divides) simplify (cut-with-single-formula "c mod modulus = c") (apply-macete-with-minor-premises mod-characterization) (unfold-single-defined-constant (0) divides) simplify)))
		(def-constant -_mod "lambda(a:zz_mod, (-a) mod modulus)" (theory arithmetic-mod-n))
	imps.mcmaster.ca /theories/reals/modular-arithmetic.html (626 words)

	Modular arithmetic - Wikipedia, the free encyclopedia
		Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations of the ring of integers: addition, subtraction, and multiplication.
		The notion of modular arithmetic is related to that of the remainder in division.
		Modular arithmetic is referenced in number theory, group theory, ring theory, abstract algebra, cryptography, computer science, and the visual and musical arts.
	en.wikipedia.org /wiki/Modular_arithmetic (1234 words)

	Modular arithmetic
		Modular arithmetic is a modified system of arithmetic for integers, sometimes referred to as 'clock arithmetic', where numbers 'wrap around' after they reach a certain value (the modulus).
		Modular arithmetic, first systematically studied by Carl Friedrich Gauss at the end of the eighteenth century, is applied in number theory, abstract algebra and cryptography.
		In abstract algebra, it is realized that modulo arithmetic is a special case of forming the factor ring of a ring modulo an ideal.
	www.ebroadcast.com.au /lookup/encyclopedia/mo/Modulo.html (756 words)

	math lessons - Advanced modular arithmetic theory
		Modular arithmetic is a system of arithmetic for integers, sometimes referred to as "clock arithmetic", where numbers "wrap around" after they reach a certain value (the modulus).
		Modular arithmetic is applied in number theory, abstract algebra, cryptography, and visual and musical art.
		In music, because of octave and enharmonic equivalency (that is, pitches in a 1/2 or 2/1 ratio are equivalent, and C# is the same as Db), modular arithmetic is used in the consideration of the twelve tone equally tempered scale, especially in twelve tone music.
	www.mathdaily.com /lessons/Advanced_modular_arithmetic_theory (1215 words)

	Modular arithmetic
		As the previous two examples show, problems arise when considering closure and inverses.
		It will actually turn out that the latter is a more serious than the former, and to sort it out we need to make a small detour into elementary number theory.
		The first fact is the well known fact about the division of integers with a quotient and remainder.
	www-history.mcs.st-and.ac.uk /~edmund/lnotes/node5.html (205 words)

	Arithmetic
		Arithmetic is the branch of mathematics dealing with integers or, more generally, numerical computation.
		Arithmetic was part of the quadrivium taught in medieval universities.
		Floating-point arithmetic is the arithmetic performed on real numbers by computers or other automated devices using a fixed number of bits.
	users.skynet.be /fa956617/math/topics/Arithmetic.html (202 words)

	Modular Arithmetic — An Introduction
		Modular arithmetic is quite a useful tool in number theory.
		Modular arithmetic lets us state these results quite precisely, and it also provides a convenient language for similar but slightly more complex statements.
		As shown by the preceding examples, one of the powers of modular arithmetic is the ability to show, often very simply, that certain equations and systems of equations have no integer solutions.
	www.math.rutgers.edu /~erowland/modulararithmetic.html (1827 words)

	MAS320, Number Theory
		The main strands are continued fractions, binary quadratic forms and modular arithmetic.
		The theory of continued fractions serves as a unifying theme as well as a source of algorithms.
		Modular arithmetic: primitive roots, quadratic residues, Legendre symbol, quadratic reciprocity.
	www.maths.qmw.ac.uk /undergraduate/modules/MAS320.html (177 words)

	Basics of Computational Number Theory
		The plan of the paper is to first give a quick overview of arithmetic in the modular integers.
		Modular arithmetic is arithmetic using integers modulo some fixed integer N.
		Modular Inverses - Given a number n and a modulus m, find the inverse of n, ie the number k such that n * k = 1 (mod m).
	www.math.umbc.edu /~campbell/NumbThy/Class/BasicNumbThy.html (2263 words)

	Number Theory at the Library of Math (Free Online Mathematics) (Site not responding. Last check: 2007-11-06)
		Roughly, number theory is the mathematical treatment of questions related to the integers; that is, the numbers 0, -1, 1,-2, 2,-3, 3...
		Gauss's many achievements in number theory are well documented; and it's he who coined the phrase "number theory is the queen of the sciences".
		The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime or a product of a finite number of primes and this factorization is unique except for the rearrangement of the factors.
	www.libraryofmath.com /Number_Theory.html (2298 words)

	Modular division via the multiplicative inverse of the denominator
		In ordinary arithmetic, the multiplicative inverse of b is the reciprocal of b, namely 1/b.
		This corresponds to the fact in ordinary arithmetic that 1 divided by 0 does not have an answer.
		The number theory to figure out such fractions is slightly more complicated than the number theory covered in this class, so we won't ask you to calculate such divisions.
	www.cs.brown.edu /courses/cs007/modmult/node2.html (549 words)

	Arithmetic, Numeration, Number Theory - Numericana
		Modular Arithmetic may be used to find the last digit(s) of very large numbers.
		This is an example of what's called modular arithmetic: The remainder in the division by some fixed modulus of a sum, a difference or a product depends only on the remainders for both operands.
		With modular arithmetic, we don't have to deal with larger and larger results because, at each iteration, we consider only the remainder of the division by m, which remains less than m.
	home.att.net /~numericana/answer/numbers.htm (7607 words)

	Modular Arithmetic, Fermat Theorem, Carmichael Numbers - Numericana
		Modular arithmetic: The algebra of congruences, formally introduced by Gauss.
		The first clean presentation of modular arithmetic was published by Carl Friedrich Gauss [ the name rhymes with house ] in
		Lagrange's Theorem (arguably the first great result of Group Theory) states that the order of any subgroup divides the order of the whole group.
	home.att.net /~numericana/answer/modular.htm (3170 words)

	Read This: An Introduction to Number Theory
		In particular, they discuss roughly four different "streams" of number theory: standard undergraduate number theory (including modular arithmetic and quadratic reciprocity, along with a small dose of basic algebraic number theory), Diophantine geometry, analytic number theory, and a little computational number theory.
		Algebra is treated in a similar fashion; when group theory would be useful in a theorem, the authors present a second proof that doesn't use it, if possible.
		Of course, a student who has never before seen ring theory or uniform convergence would be steamrollered by the speed at which those topics are reviewed; but the pace seems just right for someone who has seen them before but may not have fully internalized them.
	www.maa.org /reviews/EverestWard.html (1249 words)

	Number Theory - Mathematics and the Liberal Arts
		Boethius begins what we might think of as modular arithmetic (even and odd, and later evenly-even, evenly-odd, oddly-even), but the classification of numbers and parts of numbers soon acquires an unexpected complexity.
		Schrader, Dorothy V. The Arithmetic of the Medieval Universities.
		The applications of modular arithmetic to cryptography and fast methods of multiplication are more widely known, but will come as a pleasant surprise to the uninitiated.
	math.truman.edu /~thammond/history/NumberTheory.html (1152 words)

	Department of Mathematics - University of Georgia
		Our number theory group is complemented by a large group in algebraic geometry, including Valery Alexeev, William Graham, Elham Izadi, Roy Smith, and Robert Varley.
		Our graduate program in number theory is ranked 10th by US News and World Report.
		If you are interested in graduate studies in number theory and would like further information on our group, do not hesitate to contact any of us.
	www.math.uga.edu /research/number_theory.html (642 words)

	GRAD MTH Course Descriptions
		A study of properties of the integers, modular arithmetic.
		A study of probability theory, sample spaces, random variables, mutual exclusion, independence, conditional probability, per mutations and combinations, common discrete and continuous distributions, expected value, mean, variance, multivariate distributions, covariance, Central Limit Theorem.
		A study from the classical theory of point sets in Euclidean space and the theory of functions of one or more real variables to topology, continuous functions, and Lebesgue integral and the Henstock integral.
	www.troy.edu /catalogs/0506grad/G8MTH.htm (636 words)

	The Math Forum - Math Library - Number Theory
		In number theory, straightforward, reasonable questions are remarkably easy to ask, yet many of these questions are surprisingly difficult or even impossible to answer.
		An introduction to computational number theory, beginning with a quick overview of arithmetic in the modular integers.
		An announcement of a prize for the solution to a problem pertaining to the Diophantine equation of the form A^x + B^y = C^z where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common factor.
	mathforum.org /library/topics/number_theory (2144 words)

	Number Theory (Site not responding. Last check: 2007-11-06)
		This number theory seminar also enjoys the active participation of some of the leading figures who come to Montreal on a regular basis and give short courses suitable for graduate students.
		In recent years this included courses on Hilbert modular forms and varieties, Modular forms and the Birch-Swinnerton Dyer conjecture, Cryptography, Vector bundles on curves.
		Students specializing in number theory are expected to fulfil first the basic requirements in algebra and analysis.
	www.math.mcgill.ca /department/numtheory.php (445 words)

	Number Theory at the University of Georgia
		Arithmetic information on branched coverings, such as fields of definition.
		Special values of L-series, Fourier coefficients of metaplectic forms, analytic theory of automorphic and metaplectic forms, exponential sums, sparse polynomials and cryptography, applications of number theory to cryptography.
		Good news: our graduate program in number theory is ranked 10th by US News and World Report.
	www.math.uga.edu /~lorenz/Number_Theory_Group.html (451 words)

	Open Directory - Science: Math: Number Theory: Elliptic Curves and Modular Forms (Site not responding. Last check: 2007-11-06)
		Modular Forms and Hecke Operators - Notes by William A. Stein of a course by Ken Ribet.
		Modular Forms Course - Notes of a 1996 Berkeley course of Ken Ribet's on modular forms and Hecke operators.
		Recent Progress in the Theory of Elliptic Curves - An abstract to Henri Darmon's and Bertolini's work, which approaches a p-adic variant of the Birch - Swinnerton-Dyer conjecture, for curves of rank higher than one.
	dmoz.org /Science/Math/Number_Theory/Elliptic_Curves_and_Modular_Forms (780 words)

	math_class: Number Theory 101 (Modular Arithmetic) (Site not responding. Last check: 2007-11-06)
		Armed with this knowledge, we may wish to consider all numbers with the same remainder (when dividing by b) to be equivalent for our purposes.
		Modulo equivalence helps us simplify a bunch of arithmetic when we're only concerned about the remainder when we are done.
		Another bit of notation that is often used when doing modulo arithmetic is that one defines a % b to be a's remainder when divided by b.
	www.csh.rit.edu /~pat/math/series/nt/20020829 (1836 words)

	Modular Arithmetic Calculator (Site not responding. Last check: 2007-11-06)
		Here is a calculator for doing arithmetic (adding, subtracting, and multiplying) whole numbers on a clock with a given number of hours.
		If this doesn't make any sense to you, here's an explanation of what clock arithmetic is, with an interactive clock that shows the different names for a number on a clock in another way.
		Division is often impossible in modular arithmetic; check back here soon for an online calculator for dividing.
	www.math.csusb.edu /faculty/susan/modular/modcalc.html (360 words)

	Modular Forms and Galois Cohomology - Cambridge University Press
		This book provides a comprehensive account of a key (and perhaps the most important) theory upon which the Taylor-Wiles proof of Fermat's last theorem is based.
		The book begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and recent results on elliptic modular forms, including a substantial simplification of the Taylor-Wiles proof by Fujiwara and Diamond.
		The book will be of interest to graduate students and researchers in number theory (including algebraic and analytic number theorists) and arithmetic algebraic geometry.
	www.cambridge.org /catalogue/catalogue.asp?isbn=052177036X (241 words)

	The Math Forum - Math Library - Group Theory (Site not responding. Last check: 2007-11-06)
		Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way.
		Group theory takes an abstract approach, dealing with many mathematical systems at once and requiring only that a mathematical system obey a few simple rules, seeking then to find properties common to all systems that obey these few rules.
		A paper that presents an integrally generalized theory or conceptualization of life that includes elements of an axiomatic approach and is physically interpretable, formulated as the result of attempts to invent a holistic system of creative synthetic...more>>
	mathforum.org /library/topics/group_theory (2240 words)

	Theory of RSA Algorithm (Site not responding. Last check: 2007-11-06)
		∙Therefore, by rewriting the equation using the definition of modular arithmetic, it is proven that raising each side of a congruency to a certain power still leaves a true statement.
		∙ Rewrite the congruency statement using the definition of modular arithmetic.
		∙ Rewrite this statement using the definition of modular arithmetic.
	users.wpi.edu /~goulet/frontiers_2002/theory_of_rsa_algorithm.htm (692 words)

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