# What are some famous disagreements in math

## Two famous mathematical conjectures sparked off fire

It happens again and again that a mathematical proof turns out to be wrong. Most of the time, the public does not notice anything. But in the last few weeks two events have caused quite a stir.

Mathematics rarely makes the headlines. But in the past few weeks she has been in the spotlight twice. First, two mathematicians demystified an alleged proof of the so-called abc conjecture. A short time later, an honored mathematician announced that he had proven the 160-year-old Riemann Hypothesis in just a few pages. The mockery was not long in coming.

The abc conjecture is about triples of numbers - a, b and c - that have no prime numbers in common and for which a + b = c applies. (Prime numbers are whole numbers that can only be divided by one and by themselves.) It is believed that the product of the prime factors of these three numbers is generally greater than c. For example 6 + 25 = 31. The number 6 has the prime factors 2 and 3, 25 has the prime factor 5, and 31 is prime itself. The product of the prime factors is thus 2 × 3 × 5 × 31 = 930, which is greater than 31. (More precisely, the conjecture says that there are only finitely many such triples whose prime factor products, raised to the power of d (d> 1), are smaller are as c.)

### Basis for other theorems

The conjecture was made in the 1980s. Various attempts to prove it have failed. In the hope that this would succeed sooner or later, mathematicians have since presented several important theorems of number theory, the proofs of which are based on the correctness of the abc conjecture. However, as long as there is no conclusive evidence for the abc conjecture, these theorems cannot be considered correct either.

In 2012 a turning point appeared to be on the way. The Japanese mathematician Shinichi Mochizuki had put four papers on the Internet that, taken together, supposedly prove the abc conjecture. The papers, which amount to 500 pages and refer back to earlier publications, are extremely complex and difficult to understand. Mochizuki had specially developed a theory in arithmetic geometry, the so-called Inter-universal Teichmüller Theory, with the help of which he now tackled the proof of the abc conjecture. His international reputation - including completing his mathematics degree at Princeton University at the age of 19 as the second best - made colleagues sit up and take notice. Over the course of many months, a dozen experts subjected the four papers to a thorough examination. They found no obvious errors and tended to accept their correctness.

But not everyone was convinced. In the spring of this year, two equally renowned mathematicians flew to Japan - Peter Scholze from the University of Bonn, who was awarded a Fields Medal a few weeks ago, and Jakob Stix from the Goethe University in Frankfurt. They spent a week of intensive study with Mochizuki and his collaborator Yuichiro Hoshi. After that, they concluded that the evidence was incomplete. It is based on a dubious interim result, a so-called corollary, which is either wrong or has gaps that cannot be bridged. In an open letter they argued that the proposed proof of the abc conjecture had a serious problem and could not be salvaged by minor modifications.

One would think that there can be no disagreement in mathematics since evidence is presented in purely logical words and formulas. However, this is not the case. Mochizuki claims - and sometimes in an insulting tone - that the negative judgment of the two Germans is based on misunderstandings. They in turn replied that Mochizuki had to at least explain his theses better. A mathematician at the University of Nottingham even attributed the supposed misunderstanding to cultural differences between English-speaking and Japanese scientists. In any case, the abc conjecture cannot be considered proven as long as there are doubts about the corollary in question.

### Fanfare and disappointment

Mochizuki's alleged evidence is hundreds of pages long. A proof of the Riemann Hypothesis that the 89-year-old British mathematician Sir Michael Atiyah recently presented at the Heidelberg Laureate Forum is of a completely different format: it fits on five pages. Atiyah's lecture had been announced in advance with loud fanfare. As a result, numerous Internet portals that broadcast the 45-minute lecture live collapsed under the onslaught of curious people.

The Riemann Hypothesis was drawn up in 1859 by the German mathematician Bernhard Riemann and is considered the most famous unsolved problem in mathematics. It is important to understand the distribution of prime numbers. The assumption makes statements about the zeros of a special mathematical function, the so-called zeta function. The complex zeros of this function, so the assumption, all have the same real part, so lie in the complex plane on a straight line.

Atiyah's announcement had been received with some skepticism. But his reputation - he is the recipient of both the Fields Medal and the Abel Prize - was a guarantee of worldwide interest. The disappointment that followed was all the greater. In front of the crowded room, Atiyah said that the idea of a proof had occurred to him as a by-result of a proof for another theorem. Then he spent half an hour on historical considerations. The alleged evidence finally followed on a single graph. The credibility was not helped by the fact that Atiyah himself admitted that his work had not even been accepted for publication by the internet archive arxiv.org. He attributes the refusal to age discrimination.

### Götterdämmerung

Most mathematicians, out of sympathy for the famous professor, did not want to allow themselves to judge the presentation. Those who did anyway stated that Atiyah's presentation could not even be regarded as evidence and did not skimp on allegations to the organizers. The Heidelberg Laureate Forum allowed a well-deserved mathematician to be embarrassed in his old days.

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