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Abc ConjectureThe abc conjecture in number theory was first formulated by Joseph Oesterl and David Masser in 1985. It states that for any there exists a constant , such that for every triple of positive integers a, b, c satisfying -
we have -
where rad(n) (the radical of n) is the product of the distinct prime divisors of n. It has not been proved as of 2004. A more accurate conjecture 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 or 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. See also References - http://mathworld.wolfram.com/abcConjecture.html
- http://www.math.unicaen.fr/~nitaj/abc.html
- http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
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