Abc conjecture
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The abc conjecture in number theory was first formulated by Joseph Oesterlé and David Masser in 1985.
It states that for any <math> \varepsilon > 0 <math> there exists a constant <math> C_{\varepsilon} > 0 <math>, such that for every triple of positive integers a, b, c satisfying
- <math> a + b = c \ \mbox{and}\ \gcd(a,b) = 1 <math>
we have
- <math> c < C_{\varepsilon} \operatorname{rad}(abc)^{1+\epsilon}, <math>
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
- Greatest common divisor (gcd)