The abc conjecture in number theory was first formulated by Joseph Oesterlé 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

• Greatest common divisor (gcd)

References

• http://mathworld.wolfram.com/abcConjecture.html
• http://www.math.unicaen.fr/~nitaj/abc.html
• http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf