Maths Proof That Rocked Number Theory Will Be

might be better than usual at detecting the locations of earthquake aftershocks, MATHS PROOF THAT says Andy Frassetto, a seismologist at the Incorporated Research Institutions for ROCKED NUMBER THEORY Seismology in Washington DC. “You’ll get a signal with less noise on top, allowing you WILL BE PUBLISHED to squeeze a little more information out of those events,” he says. But some experts say author Shinichi Mochizuki The fall in noise could also benefit seismologists who use naturally occurring failed to fix fatal flaw in solution to major problem. background vibrations, such as those from crashing ocean waves, to probe Earth’s By Davide Castelvecchi divide two numbers, a and b, then only a few, crust. Because volcanic activity and large ones divide their sum, c. changes to water tables affect how fast the fter an eight-year struggle, embattled A confirmed proof could change number natural waves travel, scientists can study Japanese mathematician Shinichi theory by, for example, providing an innova- these processes by monitoring how long Mochizuki has finally received some tive approach to proving Fermat’s last theorem, it takes a wave to reach a given detector. A validation. His 600-page proof of the the legendary problem formulated by Pierre de fall in human-induced noise could boost the abc conjecture, one of the biggest Fermat in 1637 and solved only in 1994. sensitivity of detectors to natural waves at Aopen problems in number theory, has been Many mathematicians found Mochizuki’s similar frequencies, says Lecocq. accepted for publication. proof to be written in an impenetrable, idio- Belgian seismologists are not the only Acceptance of the work in Publications of the syncratic style, built entirely on unfamiliar ones to notice the effects of lockdown. Research Institute for Mathematical Sciences is mathematical concepts. Celeste Labedz, a graduate student in the latest development in a long and acrimo- Mochizuki has declined all invitations geophysics at the California Institute of nious controversy over the mathematician’s to travel abroad to lecture about his work. Technology in Pasadena, tweeted that proof. The journal, of which Mochizuki is Although, at the time, some of his close collab- a “seriously wild” fall in noise had been chief editor, is published by Japan’s Research orators said they found the proof to be correct, picked up by a station in Los Angeles. Institute for Mathematical Sciences (RIMS) at experts around the world struggled to slog But not all seismic monitoring stations Kyoto University, where he works. through it, let alone verify it. Conferences were will see such a pronounced effect, Two other RIMS mathematicians, Masaki held on the subject, and participants reported says Emily Wolin, a geologist at the Kashiwara and Akio Tamagawa, announced partial progress, but said it would probably US Geological Survey in Albuquerque, the publication at a press conference in Kyoto take many years to come to a conclusion. New Mexico. Many stations are purposefully on 3 April. The paper “will have a big impact”, In 2018, two German mathematicians — located in remote areas or deep boreholes said Kashiwara. Peter Scholze at the University of Bonn and to avoid human noise. These should see a Mochizuki, who has denied requests for Jakob Stix at Goethe University, Frankfurt — smaller decrease, or no change at all, in the interviews over the years, did not appear at privately circulated a rebuttal of the abc proof, level of high-frequency noise they record. the press conference, and did not make himself zeroing in on one crucial passage that they available to reporters. said was faulty. In September that year, the Elizabeth Gibney Eight years ago, Mochizuki posted four pair went public with their finding: an article massive papers online, claiming to have in the maths and physics magazine Quanta solved the abc conjecture. The work baffled quoted them describing a “serious, unfixable mathematicians, who spent years trying to gap”, as Stix put it. on environmental-health risks. understand it. In 2018, two respected math- In comments posted on his website at the Infection-control measures, such as social ematicians said they were confident they had time, Mochizuki brushed the criticisms aside, distancing, will probably suppress the current found a flaw in Mochizuki’s proof — something hinting that the two authors had simply failed pandemic, but the virus could return once many saw as a death blow to his claims. to understand his work. But several experts such measures are lifted. Routine wastewater The latest announcement seems unlikely to told Nature that much of the mathematics surveillance could be used as a non-invasive move many researchers over to Mochizuki’s community considered the matter settled early-warning tool to alert communities to new camp. “I think it is safe to say that there has at that point. The official acceptance of the COVID-19 infections, says Ana Maria de Roda not been much change in the community opin- papers seems unlikely to change this. Husman, an infectious-disease researcher at ion since 2018,” says Kiran Kedlaya, a number At the press conference, Tamagawa said the the Netherlands National Institute for Public theorist at the University of California, San solution itself had not changed in response to Health and the Environment in Bilthoven. The Diego, who was among the experts who put Scholze and Stix’s criticism. Some comments institute has previously monitored sewage considerable effort into trying to verify Mochi- about it will be published in the manuscript, to detect outbreaks of norovirus, antibiotic- zuki’s claimed proof. Another mathematician, but there will be no fundamental alteration, resistant bacteria, poliovirus and measles. Edward Frenkel at the University of California, said Tamagawa. de Roda Husman’s group detected traces Berkeley, says: “I will withhold my judgement In the world of mathematics, a journal’s of SARS-CoV-2 in wastewater at Schiphol air- on the publication of this work until it actually seal of approval is often not the end of the port in Tilburg, the Netherlands, only four happens, as new information might emerge.” peer-review process. An important result truly days after the country confirmed its first The abc conjecture expresses a profound becomes an accepted theorem only after the case of COVID-19 using clinical testing. The link between the addition and multiplication community has reached a consensus that it is researchers now plan to expand sampling to of integer numbers. Any integer can be fac- correct, and achieving this can take years after the capitals of all 12 provinces in the Nether- tored into prime numbers, its ‘divisors’: for a paper’s official publication. lands and 12 other sites that have not had any example, 60 = 5 × 3 × 2 × 2. The conjecture confirmed cases. roughly states that if a lot of small primes Additional reporting by David Cyranoski.