# The abc conjecture

Posted on September 4, by woit Jordan Ellenberg at Quomodocumque reports here on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin Mochizuki.

Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July August 2, On the maximal length of two sequences of integers in arithmetic progressions with the same prime divisors.

A generalization of Mason's theorem for four polynomials. A new bound for an extension of Mason's theorem for functions of several variables. Roth's theorem and the abc conjecture. The generalized Fermat equation in function fields. Number Theory 39, No. On a conjecture of Siegel.

## A meeting at Oxford

New Zealand Journal of Mathematics, vol 35, Proceedings of a symposium, Cardiff, UK, July Gaps between integers with the same prime factors. An explicit version of the theorem of Roth-Ridout.

The number of fields generated by the square root of values of a given polynomial. Number TheoryNo. Preprint [Fr2] van Frankenhuysen, Machiel A lower bound in the abc conjecture.

Contemporanea, 76, Number Theory 95no. Theorie des nombres, Semin. Diophantine problems and polynomials. Blaise Pascal 8, No. Number Theory 94, No.The abc conjecture asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed $c^{1-\varepsilon}$ for any fixed $\varepsilon \gt 0$ (if a,b,c are smooth).

ture whose age is measured in millennia, the ABC Conjecture was discovered in the rather recent and mundane year of Of course, an important open conjecture that was formed in is still a .

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Dec 17,  · The ABC conjecture has (still) not been proved. Five years ago, Cathy O'Neil laid out a perfectly cogent case for why the (at that point recent) claims by Shinichi Mochizuki should not (yet) be regarded as constituting a proof of the ABC conjecture.

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser. It is stated in terms of three positive integers, a, b and c (hence the name) that are relatively prime and satisfy a + b = c.

The abc conjecture is a conjecture due to Oesterlé and Masser in It states that, for any infinitesimal, there exists a constant such that for any three relatively prime integers,, satisfying (1.

The abc conjecture, as easy as 1, 2, 3 or not