The ABC Conjecture

Topics: Mathematics, Number theory, Prime number Pages: 2 (638 words) Published: October 19, 2013
The ABC conjecture refers to equations of the form a+b=c. It involves the concept of a square-free number: one that cannot be divided by the square of any number. Fifteen and 17 are square free-numbers, but 16 and 18 — being divisible by 42 and 32, respectively — are not.The 'square-free' part of a number n, sqp(n), is the largest square-free number that can be formed by multiplying the factors of n that are prime numbers. For instance, sqp(18)=2×3=6.

If you’ve got that, then you should get the abc conjecture. It concerns a property of the product of the three integers axbxc, or abc — or more specifically, of the square-free part of this product, which involves their distinct prime factors. It states that for integers a+b=c, the ratio of sqp(abc)r/c always has some minimum value greater than zero for any value of r greater than 1. For example, if a=3 and b=125, so that c=128, then sqp(abc)=30 and sqp(abc)2/c = 900/128. In this case, in which r=2, sqp(abc)r/c is nearly always greater than 1, and always greater than zero Shinichi MochizukiMochizuki has been working on a proof of ABC entirely by himself for nearly 20 years and has constructed his own mathematical universe and populated it with arcane terms like “inter-universal Teichmüller theory” and “alien arithmetic holomorphic structures.”The proof is spread across four long papers, each of which rests on earlier long papers.It can require a huge investment of time to understand a long and sophisticated proof, so the willingness by others to do this rests not only on the importance of the announcement but also on the track record of the authors Mochizuki’s track record certainly makes the effort worthwhile. He has proved extremely deep theorems in the past, and is very thorough in his writing, so that provides a lot of confidence And he adds that the pay-off would be more than a matter of simply verifying the claim. “The exciting aspect is not just that the conjecture may have now been solved, but that...
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