abc Conjecture

The conjecture involves triples , , of relatively prime positive integers satisfying . The conjecture was first proposed by Joseph Oesterle in 1985 and David Masser in 1988. The function (the radical) is the product of the distinct prime factors of the triple (the square-free part of ). The conjecture states that this product is "rarely" much smaller than : there are finitely many such exceptions. For a typical triple, the "quality" of the triple, , is less than 1; for exceptions, , with small. Values above 1.4 are rare; in fact, only 234 are known [2], so they are considered "high quality triples".
This conjecture is one of the most important open problems in Diophantine analysis, encoding profound connections among the prime factors of the triples as well as connections with already-proved theorems, including Fermat’s last theorem. If proved, it would imply other conjectures.

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Recently Shinichi Mochizuki of Kyoto University, claimed to have proved the conjecture [3].
References
[1] Wikipedia. "abc Conjecture." (Sept 26, 2012) en.wikipedia.org/wiki/Abc_conjecture.
[2] B. de Smit. "ABC Triples." (Sept 28, 2012) www.math.leidenuniv.nl/~desmit/abc.
[3] P. Ball, "Proof Claimed for Deep Connection between Primes," Nature [online], 2012. www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378.
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