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Topic: Abc conjecture

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	PlanetMath: ABC conjecture
		This conjecture was formulated by Masser and Oesterlé in 1980.
		The ABC conjecture is considered one of the most important unsolved problems in number theory, as many results would follow directly from this conjecture.
		This is version 14 of ABC conjecture, born on 2001-10-15, modified 2006-10-02.
	planetmath.org /encyclopedia/ABCConjecture.html (114 words)

	ABC
		ABC ALGOL is an extension of Algol 60.
		ABC is the name of an early 1980s new wave music band fronted by Martin Fry[?].
		ABC is used in cryptozoology as an acronym for alien big cat.
	www.ebroadcast.com.au /lookup/encyclopedia/ab/Abc.html (209 words)

	Conjecture
		In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has been able to prove or disprove.
		When a conjecture has been proven to be true, it becomes known as a theorem, and joins the realm of mathematical facts.
		Although many of the most famous conjectures have been tested across an astounding range of numbers, this is no guarantee against a single counterexample, which would immediately disprove the conjecture.
	www.ebroadcast.com.au /lookup/encyclopedia/co/Conjecture.html (448 words)

	ABC conjecture
		A remarkable conjecture, first put forward in 1980 by Joseph Oesterle of the University of Paris and David Masser of the Mathematics Institute of the University of Basel in Switzerland, which is now considered one of the most important unsolved problems in number theory.
		The ABC conjecture is disarmingly simple compared to most of the deep questions in number theory and, moreover, turns out to be equivalent to all the main problems that involve Diophantine equations (equations with integer coefficients and integer solutions).
		The ABC conjecture in effect translates an infinite number of Diophantine equations (including the equation of Fermat's last theorem) into a single mathematical statement.
	www.daviddarling.info /encyclopedia/A/ABC_conjecture.html (669 words)

	The abc conjecture
		Assuming the Birch and Swinnerton-Dyer conjecture, it is shown in [Go-Sz] that this conjecture is equivalent to the Szpiro conjecture for modular elliptic curves.
		[Wa2] Walsh, P.G. On a conjecture of Schinzel and Tijdeman.
		The Wieferich criterion, the ABC conjecture and Shimura's correspondence, Satya Mohit, M.Sc.
	www.math.unicaen.fr /~nitaj/abc.html (4274 words)

	Wikinfo \| Conjecture (Site not responding. Last check: 2007-10-28)
		In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove.
		Once a conjecture has been proven, it becomes known as a theorem, and it joins the realm of mathematical facts.
		The continuum hypothesis, which tries to ascertain the relative cardinality of certain infinite sets, was eventually shown to be undecidable (or independent) from the generally accepted set of axioms of set theory.
	www.wikinfo.org /wiki.php?title=Conjecture (723 words)

	Ivars Peterson's MathTrek - The Amazing ABC Conjecture (Site not responding. Last check: 2007-10-28)
		That conjecture dates back to 1955, when it was published in Japanese as a research problem by the late Yutaka Taniyama.
		The key element appears to be a problem termed the ABC conjecture, which was formulated in the mid-1980s by Joseph Oesterle of the University of Paris VI and David W. Masser of the Mathematics Institute of the University of Basel in Switzerland.
		Links between the ABC conjecture, Fermat's last theorem, and other number-theory problems are described at http://www.coe.uncc.edu/cas/flt.html, http://www.netmeg.net/faq/science/math/fermats-last-theorem/what-if-wiles-is-wro ng/, and http://www.mathsoft.com/asolve/fermat/fermat.html.
	www.maa.org /mathland/mathtrek_12_8.html (1297 words)

	UNC Charlotte Mathematics Department - What We Know About Fermat's Last Theorem
		It is a conjecture of Mazur that R = T, and it would follow from this that every lift of rho_p which ``looks modular'' (in particular the one we are interested in) is attached to a modular form.
		Conjectures arising from Diophantine approximation theory such as the ABC conjecture, the Szpiro conjecture, the Hall conjecture, etc.
		The conjecture was motivated by a theorem, due to Mason that essentially says the ABC conjecture is true for polynomials.
	www.math.uncc.edu /flt.php (3199 words)

	[No title]
		ABC conjecture +------------------------------------------------------------ ABC conjecture If a,b,c are positive integers, let N(a,b,c) be the product of the prime divisors of a,b,c, with each divisor counted only once.
		The conjecture claims that for every epsilon>0, there is a constant mu>1 such that for all coprime a,b and c=a+b, then max(a,b,c) leq mu N(a,b,c)^(1+epsilon).
		The Catalan conjecture states that every aliquot sequence This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university.
	abel.math.harvard.edu /~knill/sofia/data/number.txt (1516 words)

	Open Questions: Number Theory
		The so-called ABC conjecture, which was formulated in the mid-1980s by Joseph Oesterlé; and David Masser, is both a generalization of Fermat's Last Theorem (in the sense that it imples FLT), and also of extreme importance for Diophantine equations in general.
		The ABC conjecture is easy enough to state in elementary terms, though on the surface it has little relevance to Diophantine equations.
		The ABC conjecture enters, because if it were known to be true, then the proof of FLT would be easier, and many other Diophantine equations could be analyzed in a similar way by looking at elliptic curves and their conductors and discriminants.
	www.openquestions.com /oq-ma001.htm (4073 words)

	Science News Online, Ivars Peterson's MathTrek (12/6/97): The Amazing ABC Conjecture
		The equation of Fermat’s last theorem is one example of a type known as a Diophantine equation -- an algebraic expression of several variables whose solutions must to be rational numbers (either whole numbers or fractions, which are ratios of whole numbers).
		In contrast, the ABC conjecture states that [sqp(ABC)]^n/C does reach a minimum value if n is any number greater than 1-- even a number such as 1.0000000000001, which is just barely larger than 1.
		"The ABC conjecture is the most important unsolved problem in Diophantine analysis," Goldfeld writes in The Sciences.
	www.sciencenews.org /pages/sn_arc97/12_6_97/mathland.htm (1251 words)

	index
		There is no known evidence of prior knowledge of Beal's conjecture and all references to it begin after Beal's 1993 discovery and subsequent dissemination of it.
		While Beal's conjecture was widely received with enthusiasm by the mathematics community at large, it seems that there are always people with other motivations.
		The Beal Conjecture is sometimes referred to as "Beal's conjecture", "Beal's problem" or the "Beal problem".
	www.bealconjecture.com (857 words)

	abc Conjecture -- from Wolfram MathWorld (via CobWeb/3.1 planetlab1.cs.wisc.edu) (Site not responding. Last check: 2007-10-28)
		The abc conjecture is a conjecture due to Oesterlé and Masser in 1985.
		conjecture were true, it would imply Fermat's last theorem for sufficiently large powers (Goldfeld 1996).
		This is related to the fact that the abc conjecture implies that there are at least
	mathworld.wolfram.com.cob-web.org:8888 /abcConjecture.html (319 words)

	Abc conjecture - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab1.cs.wisc.edu) (Site not responding. Last check: 2007-10-28)
		The abc conjecture in number theory was first proposed by Joseph Oesterlé and David Masser in 1985.
		The conjecture has not been proved, but it has a large number of interesting consequences.
		While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions in number theory.
	en.wikipedia.org.cob-web.org:8888 /wiki/Abc_conjecture (436 words)

	PlanetMath: Wieferich prime
		The conjecture that only finitely many Wieferich primes exist remains unproven, though J. Silverman was able to show in 1988 that if the abc Conjecture holds, then for any positive integer
		An analysis of Wieferich primes also proved crucial to Preda Mihailescu's proof of the (formerly-named) Catalan's conjecture.
		It has been proved that the ABC conjecture implies that there are infinitely many primes that are not Wieferich (J. Silverman, "Wieferich's Criterion and the abc Conjecture", J. Number Th.
	planetmath.org /encyclopedia/WieferichPrime.html (339 words)

	AMCA: The ABC Conjecture by Kevin Broughan
		The ABC conjecture arose in a discussion between Masser and Oesterlè in 1985.
		The ABC conjecture states that if we raise the numerator to a power strictly greater than 1, then the ratio is bounded away from zero, by a constant dependent on the power (the larger the power the bigger the constant).
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/e/k/70.htm (362 words)

	Brown University Math DUG ~ The ABC Theorem for Polynomial Rings (Site not responding. Last check: 2007-10-28)
		The ABC conjecture is a number theoretical assertion about three pairwise relatively prime integers that sum to zero.
		However, many important conjectures in number theory would follow if it is true.
		One can formulate the ABC conjecture for polynomials, and here it is a theorem not a conjecture.
	math.brown.edu /dug/event.php?event=44 (128 words)

	On some polynomials allegedly related to the abc conjecture, by Alexandr Borisov (Site not responding. Last check: 2007-10-28)
		In this paper we introduce some polynomials that are probably related to the Masser-Oesterle $abc$ conjecture as they appear when one tries to follow the easy proofs of the corresponding theorem for polynomials.
		We study the distribution of their roots in usual complex and p-adic complex numbers for primes dividing $abc$.
		Using this information we prove that almost all of these polynomials (in the sense of natural density) are irreducible.
	www.math.uiuc.edu /Algebraic-Number-Theory/0060 (107 words)

	Research subjects, Nils Bruin
		Since the ABC-conjecture suggests that the complete list is finite, one may wonder if the known solutions are actually all solutions.
		The latter question is often easier to deal with, at least to bound the size of the intersection.
		The fundamental idea was used by Chabauty in 1941 to partially prove Mordell's conjecture, which was later completely proved by Faltings.
	www.cecm.sfu.ca /~nbruin/research.html (668 words)

	Power Sums (Site not responding. Last check: 2007-10-28)
		In 1772, Euler made a conjecture: If a sum of n positive kth powers equals one kth power, then n ≥ k.
		Lander, Parkin and Selfridge further conjectured that for other powers sums (k,m,n), that m+n ≥ k.
		Another power sum problem is the Fermat-Catalan conjecture, which claims there are a finite number of solutions to x
	www.maa.org /editorial/mathgames/mathgames_11_13_06.html (354 words)

	A more general abc conjecture, by Paul Vojta (Site not responding. Last check: 2007-10-28)
		This note formulates a conjecture generalizing both the abc conjecture of Masser-Oesterlé and the author's diophantine conjecture for algebraic points of bounded degree.
		It also shows that the new conjecture is implied by the earlier conjecture.
		As with most of the author's conjectures, this new conjecture stems from analogies with Nevanlinna theory; in this case it corresponds to a Second Main Theorem in Nevanlinna theory with truncated counting functions.
	www.math.uiuc.edu /Algebraic-Number-Theory/0118 (104 words)

	Good abc Triple Database Access (Site not responding. Last check: 2007-10-28)
		Query a database containing all good abc triples whose c value is less than or equal to 10
		A good abc triple is a set of 3 positive integers a, b, c such that
		A good abc example usually means a good abc triple whose ratio is greater than 1.4.
	www.phfactor.net /abc (108 words)

	[No title]
		The ABC conjecture would imply that if the prime factors of A, B, C are prescribed in advance, then there is only a finite number of solutions to the equation A + B = C (indeed it would bound C to be no more than "roughly" the product of those primes).
		So in particular there ought to be only finitely many pairs of adjacent integers whose prime factors are limited to {2, 3, 5}, or to {2, 3, 5, 7}, or whatever.
		The ABC conjecture is open, but some partial results are known; these may be among them.
	www.math.niu.edu /~rusin/known-math/99/abc_easy (1385 words)

	The Prime Page's Links++: theory/conjectures
		Goldbach's conjecture suggests that every even number greater than 2 is the sum of two primes.
		The abc conjecture - many conjectures could be proven by just proving this one difficult result.
		This page includes the abc conjecture, generalizations, consequences, tables, bibliography...
	primes.utm.edu /links/theory/conjectures (194 words)

	v7n4 (Site not responding. Last check: 2007-10-28)
		In this paper, we consider two great unproven problems in mathematics in the language of inequalities; ABC conjecture and Riemann hypothesis.
		Then we study radical function, which is contained in the heart of ABC conjecture; we find an upper bound for it by assuming Riemann hypothesis and finally by using this bound, we combine Riemann hypothesis and ABC conjecture.
		In this short note, a conjecture ([4]: J. Merikoski, Extending means of two variables to several variables, J. Ineq.
	rgmia.vu.edu.au /v7n4.html (910 words)

	MUG: abc-conjecture (12.9.96)
		Because of the help that I got in connection with my square-free question, and the related 'product' question, I am now in a position to experiment with the following programme, whose 'meaning' is simply this: one is trying to find relatively prime values of 'a' and 'b' (i.e.
		By the way, one of the very many consequences of the conjecture is Fermat's famous 'Last Theorem' (and also K.F.Roth's legendary theorem on rational approximations to algebraic numbers - for which Roth won the Fields medal).
		If any of you are able to suggest ways of improving the speed of the above procedure, I would appreciate hearing from you.
	www.math.rwth-aachen.de /mapleAnswers/html/180.html (1337 words)

	[No title] (Site not responding. Last check: 2007-10-28)
		This is the kick-off meeting of an NWO sponsored "Leraar in Onderzoek" project that will help Kennislink to take ABC to the masses.
		An ABC triple is a triple of coprime positive integers a, b, c with a + b = c and c larger than the radical of abc.
		In this talk we present an algorithm that enumerates all ABC triples with c smaller than a given upper bound N with a runtime essentially linear in N.
	www.math.leidenuniv.nl /~desmit/ic/20050909.html (202 words)

	ELLIPTIC CURVES, THE ABC CONJECTURE, AND POINTS OF SMALL CANONICAL HEIGHT (Notes From a Seminar Talk by Matt Baker) (Site not responding. Last check: 2007-10-28)
		We begin by discussing the connection between the ABC conjecture and Szpiro's conjectural relationship between the conductor and discriminant of an elliptic curve.
		Then we discuss a conjecture of Lang which predicts, in particular, that among all elliptic curves
		After giving some computational examples due to William Stein, we will then discuss a theorem of Hindry and Silverman which implies that Lang's conjecture is implied by Szpiro's (and hence by the ABC) conjecture.
	modular.fas.harvard.edu /mcs/archive/Fall2001/notes/12-10-01/12-10-01 (112 words)

	The Prime Puzzles and Problems Connection
		18.- Minimal primorial partitions (a conjecture by John Harvester)
		The first N natural numbers listed in an order such that the sum of each two adjacent of them is a prime number, and the Rivera's Algorithm
		Rivera's Conjectures about the representation of every natural number as an algebraic sum of distinct consecutive prime numbers.
	www.primepuzzles.net /conjectures (112 words)

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