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Topic: Multiplicative order

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	Order - Wikipedia, the free encyclopedia
		Order (from Latin ordo "row, rank, series, arrangement", Old French ordre from the Latin accusative, ordinem, attested in English from the 1220s).
		The word conveys a notion of "a system of parts subject to certain uniform, established ranks or proportions", an idea very central to scholastic thought, and it was used in a wide range of contexts, from architecture to angels.
		In information processing, order is a measure of the number of objects or sub-systems in a system as seen by an observer.
	en.wikipedia.org /wiki/Order (344 words)

	Encyclopedia: Order (Site not responding. Last check: 2007-10-09)
		Order may be part of the formal title of certain clubs and groups, such as in the Order of Skull and Bones.
		Order in the context of a chemical reaction is a concept of reaction kinetics, a subdiscipline of physical chemistry.
		Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering.
	www.nationmaster.com /encyclopedia/Order (1389 words)

	Order -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		Order is the opposite of (A state of lawlessness and disorder (usually resulting from a failure of government)) anarchy and ((physics) a dynamical system that is extremely sensitive to its initial conditions) chaos.
		Order of a ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group or its elements.
		Order (schema) The number of defining symbols in a (An internal representation of the world; an organization of concepts and actions that can be revised by new information about the world) schema.
	www.absoluteastronomy.com /encyclopedia/o/or/order.htm (537 words)

	[No title]
		Below a study is presented which derives all(?) panmagic squares (of prime order) with mere basic assumptions, it poses a way to parametrize those squares in a systematic manner.
		One of the ingredients is the "pan-flip transformation" which move the first row (column) below the last row (right to the column) and a reflection to move it back to the original position, this horizontal (vertical) flip is just one of the possible transformations possible in any panmagic square.
		A furthr possibility exist from order 13 on, being latin squares with rows not in linear prograssion.
	home.wanadoo.nl /aaledewinkel/Encyclopedia/p/Pan_PrimeOrder.html (1014 words)

	Math Forum - Ask Dr. Math
		The multiplicative order of any x modulo any modulus m is defined to be the smallest positive integer n such that x^n = 1 (mod m), that is, the smallest power to which x must be raised to leave a remainder of 1 when divided by m.
		Example: The multiplicative order of 2 modulo 7 is 3, because 2^1 = 2 (mod 7), 2^2 = 4 (mod 7), 2^3 = 8 = 1 (mod 7).
		Example: The multiplicative order of 2 modulo 11 is 10, because none of 2^1, 2^2,..., 2^9 are congruent to 1 (mod 11), but 2^10 = 1024 = 1193+1 = 1 (mod 11) Here are a couple of useful facts about the multiplicative* order.
	mathforum.org /library/drmath/view/55921.html (353 words)

	Understanding the Multiplicative Structure
		Problem situations in the multiplicative conceptual field (MCF) are analyzed with respect to a wide range of task variables from five structures: numeric, semantic, propositional, contextual, and mathematical.
		Additive reasoning and multiplicative or proportional reasoning can be characterized in terms of the type of compensation, additive or multiplicative, children employ for the transformation(s) applied to the problem quantities.
		Second, these mathematical principles are believed to be the foundation for multiplicative and proportional reasoning; that is, these principles are the basic theorems in actions (ala Vergnaud, 1988), from which more complex theorem in actions can be constructed by children in solving advanced multiplicative and proportional reasoning problems.
	education.umn.edu /rationalnumberproject/90_1.html (2126 words)

	APPENDIX J
		Some elements do; there is a commutative multiplicative group formed by the elements that do have inverses, and the order that group is the value of the Euler function phi(n), which equals the number of integers less than n and greater than zero that are relatively prime to n.
		The ordering of a domain D implies the cancellation law, and therefore that D is an integral domain.
		In [Coish 1959], a concept of "local ordering" was introduced for finite GF[p] which could be enough from a physical standpoint when the order of the field is made "large enough".
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (5929 words)

	Multiplicative order Info - Bored Net - Boredom (Site not responding. Last check: 2007-10-09)
		In number theory, the multiplicative order of a number a modulo n, when gcd(a,n) = 1, is the smallest integer k with
		This is the condition that the group G be cyclic.
		The condition is that the order of some a mod p for some prime p is p-1; there always are such a, and their number is in fact known, being φ(p-1).
	www.borednet.com /e/n/encyclopedia/m/mu/multiplicative_order.html (187 words)

	Lucas-Lehmer primality test - Wikipedia, the free encyclopedia
		This does not show that the multiplicative order of 11 mod 71 is 70 because some factor of 70 may also work above.
		So the multiplicative order of 11 mod 71 is 70, and thus 71 is prime.
		If a also survives the second step, then the order of a in the group (Z/nZ)* is equal to n-1, which means that the order of that group is n-1, implying that n is prime.
	en.wikipedia.org /wiki/Lucas-Lehmer_primality_test (250 words)

	Unit 7 (Site not responding. Last check: 2007-10-09)
		The maximum order mod 2^6 is 2^4 = 16, and the maximum order mod 5^4 ia phi(5^4) = 500.
		The maximum order possible is the least common multiple of these two numbers, which is 2000.
		Write functions to do addition, subtraction, multiplication by 2 (which is the same as shift left 1 place), multiplication of any two numbers (less than p, of course), and raising a number ot a power by fast exponentiation where the exponent is a 32-bit unsigned integer, all modulo p, of course.
	www2.ics.hawaii.edu /~wes/ICS623/unit7.html (1478 words)

	Index to On-Line Encyclopedia of Integer Sequences
		multiplicative order of 2 mod n, ord(2,n): A002326
		multiplicative order of x mod y, ord(x,y), sequences related to: (1) A002326 A037226 A046932 A053006 A053446 A053447 A053448 A053449 A053450 A053451 A053452 A053453
		multiplicative order of x mod y, ord(x,y), sequences related to: (2) A057764 A059499 A059885 A059886 A059887 A059888 A059889 A059890 A059891 A059892 A059907 A059908
	www.research.att.com /~njas/sequences/Sindx_Mu.html (214 words)

	Order Info - Bored Net - Boredom (Site not responding. Last check: 2007-10-09)
		Order (mathematical) of an ordered set; see also total order and total preorder
		Order (mathematical) of a group or its elements.
		Order (mathematical) of an element in some modulus, see Multiplicative order
	www.borednet.com /e/n/encyclopedia/o/or/order.html (99 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-09)
		Date: 05/12/2000 at 09:47:12 From: Jeff Olbrys Subject: Multiplicative groups of order (p-1), p = prime I have seen several passing references that multiplicative groups of order (prime-1) are cyclic, but can't find a proof.
		In Z mod 7, 3 and 5 are the generators of the multiplicative group.
		By Lagrange's theorem on the order of elements and the order of a group, h
	www.forum.swarthmore.edu /dr.math/problems/olbrys.5.12.00.html (1209 words)

	Element Operations
		Currently the Pohlig-Hellman algorithm, combined with the Shanks baby-step/giant-step and Pollard-rho algorithms, is employed, so that the time taken to compute the logarithm of an arbitrary element is usually proportional to the square root of the largest prime dividing the order of the multiplicative group of the field.
		The multiplicative order of the non-zero element a of the field F. FactoredOrder(a) : FldFinElt -> RngIntElt
		The multiplicative order of the non-zero element a of the field F as a factorization sequence.
	www.math.wisc.edu /help/magma/text372.html (1180 words)

	About money order
		A money order is a check issued by an organization such as the Postal Service or sold by third parties such as grocery stores, convenience stores, or banks and other financial services companies, to allow individuals to make payments to each other or to pay bills such as utilities and rent.
		A money order is usually considered safer for payments from unknown parties as opposed to a personal check issued by the payer, since they are generally guaranteed cashable by the receiving party, unlike traditional checks.
		-- In information processing, order is a measure of the number of objects or sub-systems in a system as seen by an observer.
	www.money-make.net /money-order.htm (315 words)

	Templatevnl_finite_int< N > class Reference
		The additive order of x is the smallest nonnegative r such that r*x == 0.
		The multiplicative order of x is the smallest r (>0) such that x^r == 1.
		Note that the multiplication group of a finite field is always cyclic.
	paine.wiau.man.ac.uk /pub/doc_vxl/core/vnl/html/classvnl__finite__int.html (654 words)

	Greg Martin - Abstract (Site not responding. Last check: 2007-10-09)
		function λ(n) is the largest multiplicative order of any integer modulo n (that is, the exponent of the group of units in the ring of integers modulo n).
		The normal order of λ(n) is known: for almost all integers n, we have log (n/λ(n)) ∼ log log n log log log n.
		In this paper, we establish the normal order of the iterated function λ(λ(n)), which can be similarly interpreted as the maximum cycle length of the “raise to the xth power modulo n” function also employed in pseudo-random number generators: we prove that log (n/λ(λ(n))) ∼ (log log n)
	www.math.ubc.ca /~gerg/papers/abstracts/NOIC.html (175 words)

	How the FFT constants were found
		Furthermore, it is the case that the multiplicative group of nonzero elements in any finite field is cyclic.
		Thus, if m is prime, the multiplicative group of nonzero elements is a cyclic group of order m-1 containing m-1.
		In order to find a generator, we need an efficient test to determine whether a given number is a generator.
	www.cis.ksu.edu /~howell/calculator/how.html (515 words)

	Second order Lambek is undecidable (Site not responding. Last check: 2007-10-09)
		Undecidability of the Second Order Lambek Calculus Max Kanovich In my previous message on the undecidability of non-commutative second order multiplicative linear logic the problem whether the second order Lambek calculus is decidable remained open.
		The point is that the Lambek calculus does not allow sequents with the empty antecedent, and, in particular, we meet problems with 1 and Weakening.
		A sequent of the form Delta, b, p^a, l_1, q^b, e - (Delta * b * l_0 * e) is derivable in the second order Lambek calculus.
	www.seas.upenn.edu /~sweirich/types/archive/1995/msg00113.html (377 words)

	Topological Transitions (Site not responding. Last check: 2007-10-09)
		The multiplicative nature of the noise and damping is a new idea and stems from the imposition of a global constraint in the theory.
		We are in the process of developing a second order stochastic algorithm which is nontrivial because of the multiplicative nature of the noise.
		The primary goal for code development during FY 2001 is to implement the second order multiplicative noise algorithm for this problem.
	t8web.lanl.gov /people/salman/nersc/topo.html (208 words)

	Multiplicative order (Site not responding. Last check: 2007-10-09)
		In number theory, the multiplicative order of anumber a modulo n, when gcd(a,n) = 1, is the smallest integer k with
		They exist exactly when there is an element of order φ(n),φ being Euler's totient function.
		When n is a prime number p, that is always the case.The condition is that the order of some a mod p for some prime p is p-1; there always aresuch a, and their number is in fact known, being φ(p-1).
	www.therfcc.org /multiplicative-order-76801.html (171 words)

	Read about Order at WorldVillage Encyclopedia. Research Order and learn about Order here! (Site not responding. Last check: 2007-10-09)
		ordinal numbers specify the order in which the items appear.
		Big O notation (mathematical), aka the order of an
		The Order (AKA Bruders Schweigen or Silent Brotherhood) was an American
	encyclopedia.worldvillage.com /s/b/Order (182 words)

	The Undecidability of Second Order Multiplicative Linear Logic - Lafont, Scedrov (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		Abstract: The multiplicative fragment of second order propositional linear logic is shown to be undecidable.
		In referring to linear logic fragments, let M stand for multiplicatives, A for additives, E for exponentials (or modalities), 1 for first order quantifiers, 2 for second order propositional quantifiers, and I for the "intuitionistic" version.
		The undecidability of second order multiplicative linear logic.
	citeseer.ist.psu.edu /lafont96undecidability.html (544 words)

	Daily Pundit Individual Archive (Site not responding. Last check: 2007-10-09)
		I don't know, but you want to make sure the multiplicative order exists before you look for it.
		Mathworld, for example, says "Multiplicate orders exist for n that are not factors of b" (b is your x) which is an untrue statement.
		All x in [1,p-1] have a multiplicative order mod p, and that order is one of the divisors of φ(p).
	www.dailypundit.com /newarchives/005277.php (354 words)

	NMBRTHRY Archives -- June 2003 (#10)
		Victor S. Miller writes: > Given an integer a (not 0 or +/- 1) we can study the average order of > a modulo n, where n varies over an interval (we can arbitrarily define > it to be 0 if a and n are not relatively prime).
		That is if the interval, is say [X,2X], the average order > should be at least something like X divided by some log X factors.
		I've found it easier to discuss the "average behavior of the index of $a$" rather than the average order of $a$.
	listserv.nodak.edu /scripts/wa.exe?A2=ind0306&L=nmbrthry&D=0&P=1030 (174 words)

	NMBRTHRY Archives -- June 2003 (#9)
		That is if the interval, is say [X,2X], the average order >should be at least something like X divided by some log X factors.
		It is certainly true that the average order over [X,2X] is at least X/(\log X)^A. In particular this follows immediately from Artin's conjecture, (and thus Hooley's proof of this conjecture under he ERH).
		Luca and I. Shparlinski, `Average multiplicative orders of elements modulo $n$', Acta Arith., 109, 2003, 387--411.
	listserv.nodak.edu /scripts/wa.exe?A2=ind0306&L=nmbrthry&D=0&P=901 (240 words)

	S.O.S. Mathematics CyberBoard :: View topic - Order of elements in the group modulo k
		Order of elements in the group modulo k
		The multiplicative group of integers modulo 2 is the trivial group, and the multiplicative group of integers modulo 4 is cyclic with order 2.
		The multiplicative group of integers modulo n is then the direct product of the multiplicative groups of these prime power factors (a.k.a.
	www.sosmath.com /CBB/viewtopic.php?p=83785&sid=6aa4ec76560d21b5afcfd538ddde7999 (738 words)

	Multiplicative order from LiveJournal (Site not responding. Last check: 2007-10-09)
		In the latter case, $v$ is (up to a multiplicative constant) the unique positive supersolution of the equation $Q^\prime (u)=0$ in $\Omega$, and one has for $Q$ an inequality of Poincar\'e type: there exists a positive continuous function...
		Potential applications of this perfect matching condition are the construction of capacity-achieving degree distributions and the determination of the number required iterations as a function of the multiplicative gap to capacity.
		It had the word multiplicative in it, and I kept saying it, and I was typing it in here, in my notes, and mulling it over.
	www.ljseek.com /search/Multiplicative%20order (688 words)

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