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The ''abc'' Conjecture in Number Theory was first proposed by Joseph Oesterlé and David Masser in 1985 . It is stated in terms of simple properties of three integers, one of which is the sum of the other two. Although there is no obvious attack on the problem, it has already become well known for the number of interesting consequences it entails. FORMULATION Let a be three Coprime Positive Integer s, and :rad(''abc''), called the Radical of ''abc'', be the Square-free product of their Distinct Prime Factor s. In other words, the product of all the unique prime factors of the three numbers, never raising a factor to a power greater than 1. The abc conjecture states that, for any ''ε'' > 0, there exists a finite ''Kε'' such that, for all coprime positive integers ''a'' + ''b'' = ''c'', : SOME CONSEQUENCES The conjecture has not been proven, but it has a large number of interesting consequences. These include both known results, and conjectures for which it gives a Conditional Proof .
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 . REFINED FORMS A stronger inequality proposed in 1996 by Alan Baker states that in the Inequality , one can replace rad(''abc'') by :ε−ωrad(''abc''), where ω is the total number of distinct primes dividing ''a'', ''b'' and ''c''. A related conjecture of Andrew Granville states that on the RHS we could also put :O(rad(''abc'') Θ(rad(''abc'')) where Θ(''n'') is the number of integers up to ''n'' divisible only by primes dividing ''n''. PARTIAL RESULTS 1986, C.L. Stewart and R. Tijdeman : : 1991, C.L. Stewart and Kunrui Yu: : 1996, C.L. Stewart and Kunrui Yu: : where ''K''1 is an absolute constant, and ''K''2 and ''K''3 are positive effectively computable constants in terms of ε. Grid-computing program In 2006, the Mathematics Department of Leiden University in the Netherlands, together with the Dutch Kennislink science institute, launched a public Grid Computing project that aims to discover so-called "a-b-c triples" which would fulfill the conjecture. The ABC@home software runs under the University Of California, Berkeley's BOINC open grid computing platform. SEE ALSO
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