# ABC conjecture and primes

The mathematical internets are abuzz with the news that the Japanese mathematician Shinichi Mochizuki has released a proof of the famous abc Conjecture.

What’s the conjecture?   Well, as we learned in preschool, we know every positive integer has a unique factorization into primes.  If n is a positive integer, let’s write rad(n) for the product of the primes which appear in the factorization, but if a prime appears more than once we use it only once.  For example rad(12)=rad(2*2*3)=6 (here * is multiplication).

Say a,b, and c are three positive numbers.  We will say that these three numbers are “neighbors” if c < rad(a*b*c).

Say a,b, and c are three positive numbers which have no primes in common in their prime factorizations.  And also say you’ve picked them so that a+b=c. Evidence suggests that almost all of these triples of numbers  are neighbors.  The abc Conjecture says that in fact there are very few such triples which fail to be neighbors:

Let $\epsilon>0$ be any fixed number (pick one as small as you like!). Then there only are finitely many triples of positive integers, a, b, and c, such that a,b,c have no common prime factors and

$c > rad(a*b*c)^{1+\varepsilon}.$

Like most number theory problems (see Fermat’s Last Theorem!), it is much easier to explain the problem then it is to find the solution.  Mochizuki has released four papers — all long and full of high powered math — which prove that the abc Conjecture is true, and does much, much more.

The reason people are excited is that the abc Conjecture is closely related to a number of other very interesting questions and conjectures in number theory.

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The abc Conjecture | OU Math Club