What is proven and unproven in life

Has the abc conjecture already been proven?

The math has a big advantage. She can make statements that are absolutely correct. In physics or astronomy, one can only make hypotheses and then either refute or verify them. But even a verified hypothesis can at some point be replaced by a new hypothesis. Our current description of gravitational force in the context of general relativity has replaced the earlier theory of Isaac Newton and will sooner or later be replaced by an even broader theory. A mathematical proposition, on the other hand, remains correct for all time - provided, of course, that its correctness can be proven mathematically exactly.

In practice, however, the matter is nowhere near as clear. Mathematicians do not always agree on what counts as "proof" and what is not. For example with this formula:

This article is included in Spectrum - The Week, 13/2017

It is the famous "abc conjecture". To understand what this is all about, one has to study the "radical" of a number (the "rad" function in the formula). It is calculated from the product of the different prime factors of a number. For example, the number 18 can be written as the product of the prime numbers 2 and 3: 18 = 2 x 3 x 3. The radical is then 2 x 3 = 6.

The abc conjecture describes the properties of three positive integers a, b and c, for which a + b = c applies and which are also coprime: There are no natural numbers other than 1, which divide all three numbers. If this is the case, then the conjecture asserts that the largest of them (c) can "hardly be greater" than the product of all the prime factors occurring in the three numbers. "Hardly bigger" means more precisely: mostly not bigger at all, and the few exceptional cases can be compensated for by suitable choice of the constants \ epsilon and d_ \ epsilon.

That all sounds very abstract and it is. The abc assumption does not affect our everyday lives. But such statements are of great importance for mathematics. This is about the properties of the numbers themselves. Proof that the abc conjecture is correct would have great implications for many other areas of mathematics (the same is true of a rebuttal of the conjecture). Because of its enormous implications for the theory of numbers, the abc conjecture has been one of the most important open questions in mathematics since 1985, when it was first established. When the Japanese Shin'ichi Mochizuki published a proof of the assumption in 2012, the response was tremendous.

However, a mathematical proof must first be checked. Every single logical step in it must be correct; every assumption must be justified because a single mistake is enough and the entire proof is invalid. This is especially true when the evidence - as in the case of Mochizuki - is hundreds of pages. Such a test takes time and one has to dig deeply to understand all the arguments and assess their validity. With Mochizuki's proof, that's exactly the problem: the method he uses is so unconventional that hardly anyone except himself understands it. Gerd Faltings, the only German recipient of the Fields Medal - the highest honor mathematicians have to award - commented on Mochizuki's proof: "I can't comment on it because I don't understand it. I don't understand the idea behind it either. He wrote 1,000 pages to spend half a year of your life with. And I don't intend to spend half a year of my life on it because I have other things to do and other obligations. "