A big event in the past two decades was the resolution of Fermat's Last Theorem. Now, an excited mathematical community is focusing on the possible resolution of another big idea...which if true, would be "one of the most astounding achievements of mathematics of the 21st century"...and would lead to an alternate proof of Fermat's Last Theorem.
On September 12, Shinichi Mochizuki, mathematician at Japan's Kyoto University, published a 500-page proof of the abc conjecture. The conjecture itself was proposed independently by mathematicians David Masser and Joseph Oesterle in 1985.
Now, the abc conjecture refers to equations of the form a+b=c. A key idea is the concept of a square-free number (i.e. quadratfrei), or a number that cannot be divided by the square of any number. For example, 13 and 21 are square free-numbers, but 12 and 18 are not, as 22 and 32 divide them respectively.
Next, we define the "square-free" part of any number n, sqp(n), to be the largest square-free number obtainable by multiplying prime factors of n. For instance, sqp(21)=3x7=21 (again showing that 21 is square-free), while sqp(12)=sqp(18)=2x3=6.
And finally, the abc conjecture refers to a special property of the square-free product of the three integers a,b, and c, that is, sqp(abc). The Conjecture states: "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." A mouthful! For example, if a=6 and b=12, then c=18 and sqp(abc)=2x3=6. And, for r=2 and r=3, sqp(abc)2/c = 12/18 and sqp(abc)3/c = 18/18. It is at this point I get lost in the implications!
So, it sounds somewhat simple...BUT...! Many outstanding mathematicians have tried to prove it using the theory of elliptic curves (as did Wiles in his proof of Fermat's Last Theorem), but they have had a difficult time teasing out the "deep connection" hidden within the prime factors of a, b and c=a+b.
What if the Conjecture is proven, or what is the gain? Dorian Goldfeld, Columbia University mathematician, responds: “The abc conjecture, if proved true, at one stroke solves many famous Diophantine problems, including Fermat's Last Theorem. If Mochizuki’s proof is correct, it will be one of the most astounding achievements of mathematics of the twenty-first century.”
So, what is the remaining difficulty? Understanding Mochizuki’s proof! Though he used elliptic equations, he apparently also introduced new mathematical techniques that "very few other mathematicians fully understand and that invoke new mathematical ‘objects’ — abstract entities analogous to more familiar examples such as geometric objects, sets, permutations, topologies and matrices.: Goldfeld adds: "At this point, he is probably the only one that knows it all."
The review process for Mochizuki’s proof will be long, challenging, and tedious. Yet, Conrad suggests that the gain is more than mere verification of a proof: "The exciting aspect is not just that the conjecture may have now been solved, but that the techniques and insights he must have had to introduce should be very powerful tools for solving future problems in number theory.”
And for the rest of us as somewhat passive observers...we can follow the making of mathematical history in progress! Yippee!
S.T. (WWU) adds this note: "For those interested in understanding a bit more about the abc conjecture...there's a nice article from the Notices from a while ago that gives some motivation for the conjecture and its connection to several big problems in Diophantine equations, particularly Fermat's Last Theorem. A link to the pdf." Be forerwarned that it requires at a rather high level of mathematical maturity (and motivation).