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Topic: Arithmetic progression

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	Arithmetic - Wikipedia, the free encyclopedia
		Arithmetic or arithmetics (from the Greek word ἀριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory.
		Primary education in mathematics often places a strong focus on arithmetic, as further studies in mathematics as well as science benefit from an understanding of arithmetic.
		The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic.
	en.wikipedia.org /wiki/Arithmetic (341 words)

	Arithmetic progression - Wikipedia, the free encyclopedia
		In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
		The sum of the components of an arithmetic progression is called an arithmetic series.
		An often-told story is that Carl Friedrich Gauss discovered it when his third grade teacher asked the class to find the sum of the first 100 numbers, and he instantly computed the answer (5050).
	en.wikipedia.org /wiki/Arithmetic_progression (285 words)

	AllRefer.com - progression (Mathematics) - Encyclopedia
		An arithmetic progression is a sequence in which each term is derived from the preceding one by adding a given number, d, called the common difference.
		An arithmetic series is the indicated sum of an arithmetic progression, and its sum of the first n terms is given by the formula [2a+(n-1)d]n/2; in the above example the arithmetic series is 3+7+11+15+…, and the sum of the first 5 terms, i.e., when n=5, is [2•3+(5-1)4] 5/2=55.
		A harmonic progression is one in which the terms are the reciprocals of the terms of an arithmetic progression; it therefore has the general form 1/a, 1/(a + d), …, 1/[a+(n-1)d].
	reference.allrefer.com /encyclopedia/P/progrsn.html (374 words)

	Grade 11 Issue 4/2004 Mathematics Magazine
		An arithmetic progression is a sequence in which each term after the first is formed by adding a fixed amount, called the common difference, to the preceding term.
		Thus to insert k arithmetic means between two numbers is to form an arithmetic progression of (k+2) terms having the two given numbers as the first and the last terms.
		A geometric progression is a sequence in which each term after the first is formed by multiplying the preceding term by a fixed number, called the common ratio.
	www.mathematicsmagazine.com /4-2004/Gr11_4_2003.htm (598 words)

	Series
		An arithmetic progression is a sequence where each term is a certain number larger than the previous term.
		In general, the nth term of an arithmetic progression, with first term a and common difference d, is: a + (n - 1)d.
		So the first term of the arithmetic progression (which is equal to the first term of the geometric progression) is either 5 or 1.
	www.mathsrevision.net /alevel/pure/series.php (909 words)

	New Page 1
		Arithmetic is the knowledge of the properties of numbers combined in arithmetic or geometric progressions.
		Or, (the sum of the first and last numbers of a progression) is twice the middle number of the progression, if the total number of numbers (in the progression) is an odd number.
		Or, (the result of multiplying the first number by the last number of a geometrical progression,) if the number of numbers (in the progression) is odd, is equal to the square of the middle number of the progression.
	www.muslimphilosophy.com /ik/Muqaddimah/Chapter6/Ch_6_19.htm (1840 words)

	Draw Sizer - Arithmetic Progression
		This utility employs arithmetic progression to compute drawer face heights by progressively adding a fixed increment to successive drawers beginning with the top drawer.
		With arithmetic progression, the heights of successive drawer faces differ by a constant amount or "increment".
		Arithmetic progression is fairly straightforward, especially when you already have values in mind for the height of the top drawer, the number of drawers, and the height increment.
	www.woodbin.com /calcs/drawsizer_arithmetic.htm (354 words)

	The Grinnell Scheme Web: Sum of an arithmetic progression (Site not responding. Last check: 2007-11-07)
		For instance, the sequence 3, 10, 17, 24, 31 is an arithmetic progression, because each term after the first is seven greater than its predecessor.
		The sum of this progression is 3 + 10 + 17 + 24 + 31, or 85.
		But in fact we only need to know three things about an arithmetic progression in order to figure out everything about it: what its first term is, what the constant difference between adjacent terms is, and how many terms there are.
	www.math.grin.edu /~stone/scheme-web/sum-of-progression.html (711 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		By default, the first value of an arithmetic progression is 0, and the increment is 1.
		Typically, the first value in a geometric progression is the base value itself, and the default base value is 2 (1 is a pretty boring geometric expression).
		A Fibonacci progression is an arithmetic progression where each value in the progression is based upon the summation of the two previous values.
	www.cs.ucd.ie /staff/rem/D011/Practical4.doc (726 words)

	The Prime Glossary: arithmetic sequence
		An arithmetic sequence (or arithmetic progression) is a sequence (finite or infinite list) of real numbers for which each term is the previous term plus a constant (called the common difference).
		In 1939, van der Corput showed that there are infinitely many triples of primes in arithmetic progression [Corput1939].
		The longest known arithmetic sequence of primes is currently of length 23, starting with the prime 56211383760397 and continuing with common difference 44546738095860, found by Markus Frind, Paul Jobling and Paul Underwood in July 2004.
	primes.utm.edu /glossary/page.php?sort=ArithmeticSequence (516 words)

	[No title]
		From: dsavitt@math.harvard.edu (David Savitt) Subject: Re: About Arithmetic Progressions Date: 12 Feb 2001 01:35:43 GMT Newsgroups: sci.math.research Summary: Sets of integers of positive density contain long arithmetic progressions freedom641@my-deja.com writes: >Let P be any finite partition of the set of all natural numbers and >let k be any fixed natural number.
		Subject: Re: About Arithmetic Progressions Date: Mon, 12 Feb 2001 15:06:32 -0600 Newsgroups: sci.math.research On 12 Feb 2001, David Savitt wrote: > freedom641@my-deja.com writes: > > >Let P be any finite partition of the set of all natural numbers and > >let k be any fixed natural number.
		Subject: Re: About Arithmetic Progressions Date: Wed, 14 Feb 2001 15:10:40 +0000 Newsgroups: sci.math.research On 12 Feb 2001, David Savitt wrote: > freedom641@my-deja.com writes: > > >Let P be any finite partition of the set of all natural numbers and > >let k be any fixed natural number.
	www.math.niu.edu /~rusin/known-math/01_incoming/szemeredi (1335 words)

	No Four Squares In Arithmetic Progression (Site not responding. Last check: 2007-11-07)
		Dickson's "History of the Theory of Numbers" presents a proof that there do not exist four squares in arithmetic progression, but it doesn't seem satisfactory.
		Thus we have four squares in arithmetic progression with the common difference 2r
		This contradicts the fact that there must be a smallest absolute common difference for four squares in arithmetic progression, so the proof is complete.
	www.mathpages.com /home/kmath044.htm (720 words)

	PlanetMath: multidimensional arithmetic progression (Site not responding. Last check: 2007-11-07)
		-dimensional arithmetic progresssion is a set of the form
		The length of the progression is defined as
		This is version 4 of multidimensional arithmetic progression, born on 2003-05-26, modified 2004-01-25.
	planetmath.org /encyclopedia/4303.html (60 words)

	Arithmetic Sequences of Perfect Squares (Site not responding. Last check: 2007-11-07)
		Then, from some facts about these integers that I get from the arithmetic progression I find two other integers, s,r, such that (s-3r), (s-r), (s+r), (s+3r) are all squares in arithmetic progression, and their common difference, 2r
		Since the right hand factors are co-prime, it follows that the four quantities (s-3r), (s-r), (s+r), (s+3r) must each have square absolute values, and they are in arithmetic progression with a common difference of 2r.
		Together these equations imply that a², b², d² and c² are in arithmetic progression, which is impossible.
	mcraefamily.com /MathHelp/PuzzleSequenceOfSquares4.htm (1028 words)

	Puzzle 262. Semiprimes in arithmetic progression
		Computing all small semi primes is easy compared to finding long arithmetic progressions among them.
		A progression of at least 25 semi primes must have difference divisible by 5# = 30 to avoid factors 2^2, 3^2 and 5^2.
		For every pair of semi primes with difference small enough to potentially extend to 30 semi primes within an x interval, array lookup was used to see how many semi primes were in arithmetic progression.
	www.primepuzzles.net /puzzles/puzz_262.htm (259 words)

	Math Trek: Progressive Primes, Science News Online, April 24, 2004 (Site not responding. Last check: 2007-11-07)
		An arithmetic progression is a sequence of numbers in which each term differs from the preceding one by the same fixed amount.
		A prime arithmetic progression is one in which the numbers are all primes.
		For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression in which primes differ by 210.
	www.sciencenews.org /articles/20040424/mathtrek.asp (747 words)

	Things of interest to number theorists
		This paper gives bounds on the index of the intersection of a Selmer group and a quotient of the dual of part of a class group in each of the two groups.
		Arithmetic and geometry of the curve 1+y^3 = x^4 (with M.J. Klassen), Acta Arithmetica, (74), 1996, 241-257.
		We will motivate this discussion with problems like finding the longest string of squares in arithmetic progression (a string of three is 1, 25, 49).
	math.scu.edu /~eschaefe/nt.html (1525 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		(a) Find the second, third, and fourth terms of the arithmetic progression with the first term -11 and difference 7.
		Find x, given that 15, x, 18 are consecutive terms of an arithmetic progression.
		Find the fourth, seventh, and ninth terms of the geometric progression with first term 2 and ratio 3.
	www.thiel.edu /mathproject/ATPS/CHPTR04/p023.htm (239 words)

	PlanetMath: arithmetic progression (Site not responding. Last check: 2007-11-07)
		The sum of terms of an arithmetic progression can be computed using Gauss's trick:
		We just add the sum with itself written backwards, and the sum of each of the columns equals to
		This is version 7 of arithmetic progression, born on 2003-05-26, modified 2004-09-24.
	planetmath.org /encyclopedia/ArithmeticProgression.html (72 words)

	arithmetic progression on Encyclopedia.com (Site not responding. Last check: 2007-11-07)
		Strategy choice in solving arithmetic word problems: are there differences between students with learning disabilities, g-v poor performance and typical achievement students?
		Arithmetic Performance of Students: Implications for Standards and Programming.
		The developmental progression of children's oral story inventions.
	www.encyclopedia.com /html/x/x-arithpr.asp (257 words)

	Primes in Arithmetic Progression Records
		This page shows the largest known case of k primes in AP (arithmetic progression) for each k, with a record history.
		Dirichlet's Theorem on Primes in Arithmetic Progressions says there are always infinitely many primes on the form c + d·n, when c and d are relatively prime - and consecutive n's are not demanded.
		The longest known AP is 23 primes in arithmetic progression by Markus Frind, Paul Jobling and Paul Underwood.
	hjem.get2net.dk /jka/math/aprecords.htm (1079 words)

	Math Trek: More Progressive Primes, Science News Online, Aug. 28, 2004 (Site not responding. Last check: 2007-11-07)
		An arithmetic progression is a sequence of numbers in which each term differs from the preceding term by the same fixed amount.
		For example, 1, 5, 9, 13, 17, and 21 is an arithmetic progression (or sequence) in which consecutive numbers differ by 4.
		For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression in which consecutive primes differ by 210.
	www.sciencenews.org /articles/20040828/mathtrek.asp (508 words)

	3-Term Arithmetic Progression
		We thus can claim that if a 9-term arithmetic progression is split into two sets, then one of the sets contains a 3-terms arithmetic progression.
		For the proof, observe that 1, 5, and 9 form a 3-term progression.
		The chosen set is the union of two subsets, one consisting of red and the other of blue points.
	www.cut-the-knot.org /Curriculum/Arithmetic/ArithmeticSequence.shtml (651 words)

	Section 6.4: Arithmetic Progressions (Site not responding. Last check: 2007-11-07)
		An arithmetic progression is a sequence of the form ak + b where a and b are fixed and k runs through integer values.
		The goal of this section is to discover whatever we can about primes in arithmetic progessions.
		Our questions about primes in arithmetic progressions can be thought of in terms of "how do the primes divide themselves among the different congruence classes modulo a?"
	www.math.mtu.edu /mathlab/COURSES/holt/dnt/special4.html (405 words)

	The Top Twenty: Consecutive Primes in Arithmetic Progression
		If a and b are relatively prime positive integers, then the arithmetic progression a, a+b, a+2b, a+3b,...
		It is conjectured that there are, but this has not even been shown in for the case of n=3 primes.
		Eleven consecutive primes in arithmetic progression require a common difference of at least 2310 and they project that a search is not feasible without a new idea or a trillion-fold improvement in computer speeds.
	primes.utm.edu /top20/page.php?id=13 (422 words)

	VisAD: Class ArithProg
		Indicates whether or not the sequence is consistent with the arithmetic progression so far.
		Indicate whether or not the value is consistent with the arithmetic progression so far.
		VisADException - The sequence isn't an arithmetic progression.
	www.ssec.wisc.edu /~dglo/visad/visad/data/in/ArithProg.html (286 words)

	id:A005115 - OEIS Search Results
		Let i, i+d, i+2d,..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; a(n) = last term, i+(n-1)d.
		In other words, smallest prime which is at the end of an arithmetic progression of n primes.
		a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.
	www.research.att.com /~njas/sequences/?Anum=A005115 (303 words)

	[No title]
		A well-known conjecture asserts that every arithmetic progression contains infinitely many integers $M$ for which $p(M)$ is odd, as well as infinitely many integers $N$ for which $p(N)$ is even (see Subbarao [22]).
		Using the theory of modular forms, we announce: \proclaim{Main Theorem 1} For any arithmetic progression $r \pmod t$, there are infinitely many integers $N\equiv r \pmod t$ for which $p(N)$ is even.
		By computing $p(n) \mod 2$ for all $n\leq 5,000,000$ we found that every arithmetic progression with modulus $t\leq 100,000$ contains an integer $M$ for which $p(M)$ is odd.
	www.maths.tcd.ie /EMIS/journals/ERA-AMS/1995-01-005/1995-01-005.tex.html (1354 words)

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