A Beijing-based online education start-up has developed an artificial intelligence-powered maths app that can check children's arithmetic problems through the simple snap of a photo. Based on the image and its internal database, the app automatically checks whether the answers are right or wrong. From a report: Known as Xiaoyuan Kousuan, the free app launched by the Tencent Holdings-backed online education firm Yuanfudao, has gained increasing popularity in China since its launch a year ago and claims to have checked an average of 70 million arithmetic problems per day, saving users around 40,000 hours of time in total. Yuanfudao is also trying to build the country's biggest education-related database generated from the everyday experiences of real students. Using this, the six-year-old company -- which has a long line of big-name investors including Warburg Pincus, IDG Capital and Matrix Partners China -- aims to reinvent how children are taught in China. "By checking nearly 100 million problems every day, we have developed a deep understanding of the kind of mistakes students make when facing certain problems," said Li Xin, co-founder of Yuanfudao -- which means "ape tutor" in Chinese -- in a recent interview. "The data gathered through the app can serve as a pillar for us to provide better online education courses."

According to a new study published in the journal Science Advances, researchers from Australia and France have shown that bees can perform simple arithmetic, adding and subtracting small numbers by studying color-coded shapes. CNET reports: To test the buzzers' ability to perform arithmetic, the team used a three-chambered maze shaped like a Y, training bees to enter through a hole into a small chamber where they would see their first stimulus: blue or yellow shapes on a plain, grey background. The number of shapes varied between 1 and 5 and the color of the shapes told the bee whether it needed to add one (blue) or subtract one (yellow) from the initial number. The bee then flew into a subsequent chamber which presented both a correct option and an incorrect option. To train the bees, the correct option rewarded the critters with a drop of tasty sugar solution -- a delightful dessert for the bee. On the other hand, selecting the incorrect solution resulted in a nasty drop of quinine -- like a slab of Brussels sprouts slathered in chocolate.

The testing procedure itself focused on 14 bees undergoing four tests of 10 choices. The tests themselves were "non-reinforced," so they didn't receive reward or punishment when selecting their "answers" during testing. Because the bees were subjected to two answers each time, the expectation is that -- purely by chance -- they would select the correct answer 50 percent of the time. But the bees performed significantly better than chance would predict, selecting the correct answer around 65 percent of the time.

Tesla CEO Elon Musk announced today that the company would cut 7% of its workforce in order to cut costs as the company prepares to ramp up production and boost margins as they get closer to releasing the long-awaited $35,000 version of the Model 3. CNBC reports: Musk says Tesla faces "an extremely difficult challenge" in making their products a competitive alternative to traditional vehicles, adding that he expects Q4 profit to come in significantly lower than Q3. Five experts weigh in on whether it's a challenge Musk and Tesla can overcome:

- Oppenheimer managing director Colin Rusch agrees with Jed Dorsheimer on Tesla's job cuts, but isn't bullish on what they'll accomplish.
- Canaccord Genuity's Jed Dorsheimer thinks the workforce cut is just fine, calling it "clean-up" after the company's latest push to ramp up Model 3 production came with a wealth of new hires.
- "They're certainly in a better position than they were eight or nine months ago," says ROTH Capital's Craig Irwin. "Where we're going to see pressure on the stock today is the 'copy-paste' expectations of Q3 going through 2019 need to be reset."
- Needham's Raji Gil thinks that Tesla may have overestimated how many people can actually afford a high-end electric vehicle. "Clearly, in my mind, they have an issue with demand," says Rusch, " If you do the math, you have to conclude that 90 percent of the reservations that have been built up over the past couple of years are folks that wanted the standard battery version of the vehicle, which is $35,000."
- Westly Group founder Steve Westly loves where Elon Musk's company is right now, calling Tesla "the iPhone of electric vehicles," and saying they're well ahead of the game when it comes to a quickly-changing auto market.

Joshua S. Goldstein, a professor emeritus of international relations at American University, and Staffan A. Qvist, an energy engineer and consultant, writing for The Wall Street Journal: Climate scientists tell us that the world must drastically cut its fossil fuel use in the next 30 years to stave off a potentially catastrophic tipping point for the planet. Confronting this challenge is a moral issue, but it's also a math problem -- and a big part of the solution has to be nuclear power. Today, more than 80% of the world's energy comes from fossil fuels, which are used to generate electricity, to heat buildings and to power car and airplane engines. Worse for the planet, the consumption of fossil fuels is growing quickly as poorer countries climb out of poverty and increase their energy use. Improving energy efficiency can reduce some of the burden, but it's not nearly enough to offset growing demand.

Any serious effort to decarbonize the world economy will require, then, a great deal more clean energy, on the order of 100 trillion kilowatt-hours per year, by our calculations -- roughly equivalent to today's entire annual fossil-fuel usage. A key variable is speed. To reach the target within three decades, the world would have to add about 3.3 trillion more kilowatt-hours of clean energy every year. Solar and wind power alone can't scale up fast enough to generate the vast amounts of electricity that will be needed by midcentury, especially as we convert car engines and the like from fossil fuels to carbon-free energy sources. Even Germany's concerted recent effort to add renewables -- the most ambitious national effort so far -- was nowhere near fast enough. A global increase in renewables at a rate matching Germany's peak success would add about 0.7 trillion kilowatt-hours of clean electricity every year. That's just over a fifth of the necessary 3.3 trillion annual target.

"One of the world's foremost mathematicians, Prof Sir Michael Atiyah, has died at the age of 89," reports the BBC.

"He has been described to me by more than one professor of mathematics as the best mathematician in this country since Sir Isaac Newton," his brother tells the BBC. Slashdot reader OneHundredAndTen shared their report: Sir Michael was best known for his co-development of a branch of mathematics called topological K-theory and the Atiyah-Singer index theorem. His research also involved deep insights relating to mathematical concepts known as "vector bundles". His work in these areas has helped theoretical physicists to advance their understanding of quantum field theory and general relativity.
In September, Atiyah also claimed to have proved the 160-year-old Riemann hypothesis.

"If the hypothesis is proven to be correct," New Scientist reported, "mathematicians would be armed with a map to the location of all such prime numbers, a breakthrough with far-reaching repercussions in the field."

A federal district court has ruled that the state of Oregon illegally infringed on a man's First Amendment rights for fining him $500 because he wrote "I am an engineer" in a 2014 email to the state's Engineering Board. The court ruled that the provision in the law he broke is unconstitutional, which opens the door for people in the state to legally call themselves "engineers." Motherboard reports: This dystopian saga dates back to 2013, when Mats Jarlstrom's wife, while driving, was caught by a red light camera near their home in Beaverton, Oregon. Rather than pay the red light camera fine, Jarlstrom, an electrical engineer, spent months researching the specifics of yellow light timing and red light cameras, and learned that his wife had likely been ticketed for running a yellow light. Jarlstrom began sharing his findings on his personal website, at conferences, and even got featured on 60 Minutes. He also wrote several emails to the Oregon Board of Engineers explaining what he had found. In the email, he noted that he was an "engineer."

Rather than looking into whether traffic light timing should be changed, however, the board sent Jarlstrom a warning -- and then a $500 fine for the crime of "practicing engineering without being registered." Jarlstrom had violated one of Oregon's "Title Laws," which states that "no persons may ... hold themselves out as an 'engineer'" unless they are an "individual who is registered in this state and holds a valid certificate to practice engineering in this state." Jarlstrom has a bachelor's degree in electrical engineering and spent his career working in electronics, but wasn't board certified. He sued the state's engineering board and, last week, a U.S. District Court judge for the District of Oregon ruled that the state's law is unconstitutional. The judge wrote: "The statutes prohibit truthfully describing oneself as an 'engineer,' in any context. This restriction clearly controls and suppresses protected speech, and enforcement of the statute against protected speech is not a hypothetical threat. The term 'engineer,' standing alone, is neither actually nor inherently misleading. Courts have long recognized that the term 'engineer' has a generic meaning separate from 'professional engineer' and that the term has enjoyed 'widespread usage in job titles in our society to describe positions which require no professional training.'"

"The judge ordered that the word 'engineer' be struck from Oregon's law, which is 'substantially overbroad in violation of the First Amendment' and specifically noted that Jarlstrom may describe himself publicly and privately using the word 'engineer' and that he may continue to talk about traffic light timing publicly," reports Motherboard.

chalsall (Slashdot reader #185), writes: The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^82,589,933-1, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018.

GIMPS has been on amazing lucky streak, finding triple the expected number of new Mersenne primes -- a dozen in the last fifteen years.
"This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes is incorrect," notes GIMPS. "However, if the trend continues it may be worth further investigation. " They also report that the newly-discovered prime number "is more than one and a half million digits larger than the previous record prime number" -- and it's one of just 51 known Mersenne prime numbers ever discovered. "GIMPS, founded in 1996, has discovered the last 17..."

Patrick Laroche is one of thousands of volunteers using GIMPS' free software to hunt for prime numbers -- and is now eligible for a $3,000 "research discovery award," the group writes at mersenne.org. "GIMPS' next major goal is to win the $150,000 award administered by the Electronic Frontier Foundation offered for finding a 100 million digit prime number" -- of which $50,000 will be awarded to the discoverer, with another $50,000 going to a 501(c)(3) mathematics-related charity selected by GIMPS, and $50,000 retained by GIMPS to cover expenses and fund other awards.

dmoberhaus shares a report from Motherboard: A team of Japanese researchers from Keio University in Tokyo have demonstrated that an amoeba is capable of generating approximate solutions to a remarkably difficult math problem known as the "traveling salesman problem." The traveling salesman problem goes like this: Given an arbitrary number of cities and the distances between them, what is the shortest route a salesman can take that visits each city and returns to the salesman's city of origin. As these Japanese researchers demonstrated, a certain type of amoeba can be used to calculate nearly optimal solutions to the traveling salesman problem for up to eight cities. Even more remarkably, the amount of time it takes the amoeba to reach these nearly optimal solutions grows linearly, even though the number of possible solutions increases exponentially. The reason this amoeba is considered especially useful in biological computing is because it can extend various regions of its body to find the most efficient way to a food source and hates light.

To turn this natural feeding mechanism into a computer, the Japanese researcher placed the amoeba on a special plate that had 64 channels that it could extend its body into. This plate is then placed on top of a nutrient rich medium. The amoeba tries to extend its body to cover as much of the plate as possible and soak up the nutrients. Yet each channel in the plate can be illuminated, which causes the light-averse amoeba to retract from that channel. To model the traveling salesman problem, each of the 64 channels on the plate was assigned a city code between A and H, in addition to a number from 1 to 8 that indicates the order of the cities. To guide the amoeba toward a solution to the traveling salesman problem, the researchers used a neural network that would incorporate data about the amoeba's current position and distance between the cities to light up certain channels. The neural network was designed such that cities with greater distances between them are more likely to be illuminated than channels that are not. When the algorithm manipulates the chip that the amoeba is on it is basically coaxing it into taking forms that represent approximate solutions to the traveling salesman problem.

An anonymous reader shares a report: One day this fall, Ashutosh Garg, the chief executive of a recruiting service called Eightfold.ai, turned up a resume that piqued his interest. It belonged to a prospective data scientist, someone who unearths patterns in data to help businesses make decisions, like how to target ads. But curiously, the resume featured the term "data science" nowhere.

Instead, the resume belonged to an analyst at Barclays who had done graduate work in physics at the University of California, Los Angeles. Though his profile on the social network LinkedIn indicated that he had never worked as a data scientist, Eightfold's software flagged him as a good fit. He was similar in certain key ways, like his math and computer chops, to four actual data scientists whom Mr. Garg had instructed the software to consider as a model.

The idea is not to focus on job titles, but "what skills they have," Mr. Garg said. "You're really looking for people who have not done it, but can do it." The power of such technology will be immediately apparent to any employer scrambling to fill jobs in a tight labor market -- not least positions for data scientists, whom companies like Google, Facebook and Amazon are competing to attract. Thanks to services like Eightfold, which rely on sophisticated algorithms to match workers and jobs, many employers may soon have access to a universe of prospective workers -- even for hard-to-fill roles -- whom they might not otherwise have come across.

Paradoxically, the abundance of tight interactions among living species usually leads to disasters in ecological models. New analyses hint at how nature seemingly defies the math. Veronique Greenwood, writing for Quantamagazine: Behind the beautiful facade of a rainforest, a savanna or a placid lake is a world teeming with contests and partnerships. Species are competing for space, consuming one another for resources, taking advantage of one another's talents, and brokering trades of nutrients. But there's something funny about this picture. When ecologists try to model ecosystems using math, they tend to find that the more interactions there are among species, the more unstable the system. For a simple ecosystem model to be stable, all the interactions among its species must be in perfect harmony. Maintaining that balancing act gets much harder, however, as the number of coupled species and the strengths of their interactions rise: Any disturbance or imbalance for one couple ripples outward and sows chaos throughout the network.

Bring in mutualisms, relationships in which species contribute directly to each other's survival, and things can really fly off the handle. Pairs of organisms that live off each other sometimes do so well in the mathematical simulations -- thriving exponentially in extreme cases, in what Robert May, the theoretical ecology pioneer, once called "an orgy of mutual benefaction" -- that everything else can go extinct. It seems unlikely that real ecosystems are quite this flimsy. In a new paper in Nature Communications, a pair of theoretical ecologists at the University of Illinois explored more precisely how the give-and-take in mutualism affects ecosystem stability and how, under the right conditions, it might contribute to it. Their result joins previous work in suggesting how real-world communities manage to be more resilient than the models imply.

A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years. Erica Klarreich, writing for Quanta Magazine: On September 16, 2011, an anime fan posted a math question to the online bulletin board 4chan about the cult classic television series The Melancholy of Haruhi Suzumiya . Season one of the show, which involves time travel, had originally aired in nonchronological order, and a re-broadcast and a DVD version had each further rearranged the episodes. Fans were arguing online about the best order to watch the episodes, and the 4chan poster wondered: If viewers wanted to see the series in every possible order, what is the shortest list of episodes they'd have to watch? In less than an hour, an anonymous person offered an answer -- not a complete solution, but a lower bound on the number of episodes required. The argument, which covered series with any number of episodes, showed that for the 14-episode first season of Haruhi, viewers would have to watch at least 93,884,313,611 episodes to see all possible orderings. "Please look over [the proof] for any loopholes I might have missed," the anonymous poster wrote.

The proof slipped under the radar of the mathematics community for seven years -- apparently only one professional mathematician spotted it at the time, and he didn't check it carefully. But in a plot twist last month, the Australian science fiction novelist Greg Egan proved a new upper bound on the number of episodes required. Egan's discovery renewed interest in the problem and drew attention to the lower bound posted anonymously in 2011. Both proofs are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years. Mathematicians quickly verified Egan's upper bound, which, like the lower bound, applies to series of any length. Then Robin Houston, a mathematician at the data visualization firm Kiln, and Jay Pantone of Marquette University in Milwaukee independently verified the work of the anonymous 4chan poster. Now, Houston and Pantone, joined by Vince Vatter of the University of Florida in Gainesville, have written up the formal argument. In their paper, they list the first author as "Anonymous 4chan Poster."

An anonymous reader quotes a report from IEEE Spectrum: Deep learning has a DRAM problem. Systems designed to do difficult things in real time, such as telling a cat from a kid in a car's backup camera video stream, are continuously shuttling the data that makes up the neural network's guts from memory to the processor. The problem, according to startup Flex Logix, isn't a lack of storage for that data; it's a lack of bandwidth between the processor and memory. Some systems need four or even eight DRAM chips to sling the 100s of gigabits to the processor, which adds a lot of space and consumes considerable power. Flex Logix says that the interconnect technology and tile-based architecture it developed for reconfigurable chips will lead to AI systems that need the bandwidth of only a single DRAM chip and consume one-tenth the power.

Mountain View-based Flex Logix had started to commercialize a new architecture for embedded field programmable gate arrays (eFPGAs). But after some exploration, one of the founders, Cheng C. Wang, realized the technology could speed neural networks. A neural network is made up of connections and "weights" that denote how strong those connections are. A good AI chip needs two things, explains the other founder Geoff Tate. One is a lot of circuits that do the critical "inferencing" computation, called multiply and accumulate. "But what's even harder is that you have to be very good at bringing in all these weights, so that the multipliers always have the data they need in order to do the math that's required. [Wang] realized that the technology that we have in the interconnect of our FPGA, he could adapt to make an architecture that was extremely good at loading weights rapidly and efficiently, giving high performance and low power."

An anonymous reader shares a report: Factorising numbers into their constituent primes may sound esoteric, but the one-way nature of the problem -- and of some other, closely related mathematical tasks -- is the foundation on which much modern encryption rests. Such encryption has plenty of uses. It defends state secrets, and the corporate sort. It protects financial flows and medical records. And it makes the $2trn e-commerce industry possible. Nobody, however, is certain that the foundation of all this is sound. Though mathematicians have found no quick way to solve the prime-factors problem, neither have they proved that there isn't one. In theory, any of the world's millions of professional or amateur mathematicians could have a stroke of inspiration tomorrow and publish a formula that unravels internet cryptography -- and most internet commerce with it.

In fact, something like this has already happened. In 1994 Peter Shor, a mathematician then working at Bell Laboratories, in America, came up with a quick and efficient way to find a number's prime factors. The only catch was that for large numbers his method -- dubbed Shor's algorithm -- needs a quantum computer to work. Quantum computers rely on the famous weirdness of quantum mechanics to perform certain sorts of calculation far faster than any conceivable classical machine. Their fundamental unit is the "qubit", a quantum analogue of the ones and zeros that classical machines manipulate. By exploiting the quantum-mechanical phenomena of superposition and entanglement, quantum computers can perform some forms of mathematics -- though only some -- far faster than any conceivable classical machine, no matter how beefy.

In a paper published Thursday in the journal Science, Dr. Sergey Bravyi and his team reveal that they've developed a mathematical proof which, in specific cases, illustrates the quantum algorithm's inherent computational advantages over classical. Engadget reports: "It's good to know, because results like this become parts of algorithms," Bob Sutor, vice president of IBM Q Strategy and Ecosystem, told Engadget. "They become part of decisions about how people will start to attack problems. Where will they try classical techniques? Where will they try quantum techniques? How will those interplay? How will they work back and forth together?" What's more, the proof shows that, in these cases, the quantum algorithm can solve the problem in a fixed number of steps, regardless of how many inputs are added. With a classical computer, the more inputs you add, the more steps it needs to take in order to solve. Such are the advantages of parallel processing.

"The main point of this paper is not that somehow we discover some incredibly important quantum algorithm, or some practical, interesting problem," Bravyi told Engadget. "We ask if we can separate a constant depth [between] quantum and classical algorithms. As we increase the problem size, the runtime of the quantum algorithm remains constant, but the total number of operations grows." As Bravyi points out, this new proof doesn't, in and of itself, solve any existing computational issues. Instead, "it gives us insight into what makes a quantum computers more powerful," he continued. "And hopefully in the future it will lead to more practical, useful algorithms."

A new study in Frontiers in Psychology examined why people struggle so much to solve statistical problems, particularly why we show a marked preference for complicated solutions over simpler, more intuitive ones. Chalk it up to our resistance to change. From a report: The study concluded that fixed mindsets are to blame: we tend to stick with the familiar methods we learned in school, blinding us to the existence of a simpler solution. Roughly 96 percent of the general population struggles with solving problems relating to statistics and probability. Yet being a well-informed citizen in the 21st century requires us to be able to engage competently with these kinds of tasks, even if we don't encounter them in a professional setting. "As soon as you pick up a newspaper, you're confronted with so many numbers and statistics that you need to interpret correctly," says co-author Patrick Weber, a graduate student in math education at the University of Regensburg in Germany. Most of us fall far short of the mark.

Part of the problem is the counterintuitive way in which such problems are typically presented. Meadows presented his evidence in the so-called "natural frequency format" (for example, 1 in 10 people), rather than in terms of a percentage (10 percent of the population). That was a smart decision, since 1-in-10 a more intuitive, jury-friendly approach. Recent studies have shown that performance rates on many statistical tasks increased from four percent to 24 percent when the problems were presented using the natural frequency format.

Economist Paul Romer, a co-winner of the 2018 Nobel Prize in economics, uses the programming language Python for his research, according to Quartz. Romer reportedly tried using Wolfram Mathematica to make his work transparent, but it didn't work so he converted to a Jupyter notebook instead. From the report: Romer believes in making research transparent. He argues that openness and clarity about methodology is important for scientific research to gain trust. As Romer explained in an April 2018 blog post, in an effort to make his own work transparent, he tried to use Mathematica to share one of his studies in a way that anyone could explore every detail of his data and methods. It didn't work. He says that Mathematica's owner, Wolfram Research, made it too difficult to share his work in a way that didn't require other people to use the proprietary software, too. Readers also could not see all of the code he used for his equations.

Instead of using Mathematica, Romer discovered that he could use a Jupyter notebook for sharing his research. Jupyter notebooks are web applications that allow programmers and researchers to share documents that include code, charts, equations, and data. Jupyter notebooks allow for code written in dozens of programming languages. For his research, Romer used Python -- the most popular language for data science and statistics. Importantly, unlike notebooks made from Mathematica, Jupyter notebooks are open source, which means that anyone can look at all of the code that created them. This allows for truly transparent research. In a compelling story for The Atlantic, James Somers argued that Jupyter notebooks may replace the traditional research paper typically shared as a PDF.

Cloudflare, which is celebrating its eighth birthday has announced yet another service: an at-cost domain registrar. From a report: While Cloudflare had already been handling domain registration through the company's Enterprise Registrar service, that service was intended for some of Cloudflare's high-end customers who wanted extra levels of security for their domain names. The new domain registrar business -- called Cloudflare Registrar -- will eventually be open to anyone, and it will charge exactly what it costs for Cloudflare to register a domain. As Cloudflare CEO Matthew Prince wrote in a blog post this week, "We promise to never charge you anything more than the wholesale price each TLD charges." That includes the small fee assessed by ICANN for each registration.

Prince said that he was motivated to take the company into the registrar business because of Cloudflare's own experience with registrars and by the perception that many registrars are in the business mostly to up-sell things that require no additional effort. "All the registrar does is record you as the owner of a particular domain," Prince said. "That just involves sending some commands to an API. In other words, domain registrars are charging you for being a middle-man and delivering essentially no value to justify their markup." Charging overhead for that sort of service, Prince said, "seemed as nutty to us as certificate authorities charging to run a bit of math." (Cloudflare also provides free SSL certificates.)

A study of school grades of more than 1.6 million students shows that girls and boys perform similarly in science, technology, engineering and math (STEM) subjects. From a report: The research, published today in Nature Communications, also shows that girls do better than boys in non-STEM subjects. Our results provide evidence that large gaps in the representation of women in STEM careers later in life are not due to differences in academic performance. One explanation for gender imbalance in STEM is the "variability hypothesis." This is the idea that gender gaps are much larger at the tails of the distribution -- among the highest and lowest performers -- than in the middle.

Slashdot reader OneHundredAndTen writes: Sir Michael Atiyah claims to have proved the Riemann hypothesis. This is not some internet crank, but one the towering figures of mathematics in the second half of the 20th century. The thing is, he's almost 90 years old. According to New Scientist, Atiyah is set to present his "simple proof" of the Riemann hypothesis on Monday at the Heidelberg Laureate Forum in Germany. Atiyah has received two awards often referred to as the Nobel prizes of mathematics, the Fields medal and the Abel Prize; he also served as president of the London Mathematical Society, the Royal Society and the Royal Society of Edinburgh.

"[T]he hypothesis is intimately connected to the distribution of prime numbers, those indivisible by any whole number other than themselves and one," reports New Scientist. "If the hypothesis is proven to be correct, mathematicians would be armed with a map to the location of all such prime numbers, a breakthrough with far-reaching repercussions in the field."

Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years. Quanta Magazine: In a report [PDF] posted online Thursday, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a "serious, unfixable gap" within a mammoth series of papers by Shinichi Mochizuki, a mathematician at Kyoto University who is renowned for his brilliance. Posted online in 2012, Mochizuki's papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory. Despite multiple conferences dedicated to explicating Mochizuki's proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford University, has called "a sense of infinite regress."

Between 12 and 18 mathematicians who have studied the proof in depth believe it is correct, wrote Ivan Fesenko of the University of Nottingham in an email. But only mathematicians in "Mochizuki's orbit" have vouched for the proof's correctness, Conrad commented in a blog discussion last December. "There is nobody else out there who has been willing to say even off the record that they are confident the proof is complete." Nevertheless, wrote Frank Calegari of the University of Chicago in a December blog post, "mathematicians are very loath to claim that there is a problem with Mochizuki's argument because they can't point to any definitive error." That has now changed. In their report, Scholze and Stix argue that a line of reasoning near the end of the proof of "Corollary 3.12" in Mochizuki's third of four papers is fundamentally flawed. The corollary is central to Mochizuki's proposed abc proof. "I think the abc conjecture is still open," Scholze said. "Anybody has a chance of proving it."