Prime numbers are the building blocks of mathematics, and yet their distribution is a mystery that has stumped mathematicians for centuries. We still don’t understand why some primes sit so far apart from the next and some so close together. James Maynard is a mathematician working primarily in the world of prime numbers. In 2022 he was awarded the Fields medal, the most prestigious award in maths for those under 40, for his contributions to analytic number theory.

In this video, James joins 快猫短视频 reporter Alex Wilkins to talk about why primes are so fascinating, his award-winning career and how he thinks AI will change mathematics.

Read more: Why prime numbers might not be random after all

The seemingly random distribution of prime numbers has confounded some of the best mathematical minds for centuries. But the Riemann hypothesis, which relates to the zeros in a mathematical function, may hold the answer. It appears to show exactly where we can expect a prime number to appear, the only problem is, no one has yet been able to prove the hypothesis.

In this video we’ll explore prime numbers, explain the enigmatic zeta function and show how this mathematical proof may reveal a deeper truth about the universe.

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After a six-year drought, we now have a new largest known prime number, thanks to an amateur mathematics sleuth who deployed an army of graphics processing units (GPUs) to crunch through the possibilities.

Prime numbers are those divisible only by 1 and themselves, such as 2, 3 and 5.聽 There are an infinite number of primes, but proving which numbers are actually prime becomes harder the larger they get. We can now add 2^136,279,841-1 to the list, which at long is the biggest prime number currently known.

It was discovered by a relatively new member of a group called the (GIMPS), where thousands of people have downloaded software to hunt for prime numbers. Those lucky enough to discover one earn a place in prime number history, but also a $3000 prize. This is the first prize to be awarded since 2018.

The new prime number, labelled by the GIMPS group as M136279841, was found by , who formerly worked for Nvidia as an engineer developing GPUs, and has been searching for big primes for just under a year.

All previous GIMPS discoveries were made by computer CPUs in relatively humble personal computers, but Durant鈥檚 past at Nvidia exposed him to GPUs 鈥� the chips originally designed for powering computer games but also key to the recent rise in AI computing. He believed they would be ideal for hunting prime numbers and took advantage of a GPU system for its number-crunching abilities. He networked thousands of GPUs housed in 24 data centres across 17 countries, and has been described by the GIMPS project as a 鈥減rolific contributor鈥�.

鈥淚t was a pretty big surprise, but I had been working hard to grow the system, so stayed aware of a relatively decent chance,鈥� says Durant. 鈥淚 joined for a lot of reasons, in part to learn more about big math and information, show GPU capabilities at traditional computing, and support some tremendous software and technology developed by the GIMPS community.鈥�

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The new prime is the 52nd of a specific type called Mersenne primes to ever be discovered. Named for the French monk and mathematician Marin Mersenne, these primes are exactly one less than a power of two 鈥� which makes them slightly easier to find, and therefore the focus of GIMPS.

at Imperial College London says there is absolutely no practical application for the finding, but that the same can initially be said for lots of mathematical research. 鈥淭here鈥檚 no use for extremely large prime numbers now, but it鈥檚 not at all inconceivable that one day somebody will find something,鈥� says Buzzard. 鈥淎nd then they鈥檒l look at the maths research community and say, 鈥楽o, where are your very large prime numbers?鈥� and they鈥檒l say, 鈥榃ell, actually, we鈥檝e been thinking about that for decades鈥︹��.鈥�

After a saga eight years in the making, a mathematician is finally set to formally publish a proof that rocked number theory and baffled almost everyone who read it 鈥� including other mathematicians.

In 2012, Shinichi Mochizuki at Kyoto University in Japan produced a massive proof claiming to have solved a long-standing problem called the .

Spanning 500 pages across four papers, Mochizuki鈥檚 proof was written in an impenetrable style, and number theorists its underlying ideas.

The work has finally been accepted in the peer-reviewed journal Publications of the Research Institute for Mathematical Sciences, but a publication date hasn鈥檛 been decided yet.

Mochizuki himself is the editor-in-chief of the journal, which is also published by Kyoto University. He hasn鈥檛 been involved in the decision to publish the proof, .

First proposed in the 1980s, the ABC conjecture is based around the equation a + b = c, and concerns the link between the addition and multiplication of integers, or whole numbers.

Simply put, it says that if a and b are made up of large powers of prime numbers 鈥� numbers only divisible by themselves and one 鈥� then c isn鈥檛 usually divisible by large powers of primes.

Mathematicians have long believed that the conjecture was true, but nobody had ever been able to prove it. Mochizuki grappled with the conjecture by developing a new type of mathematics called inter-universal Teichm眉ller theory.

, mathematicians Peter Scholze at the University of Bonn in Germany and Jakob Stix at Goethe University in Germany said that they had found a 鈥渟erious, unfixable gap鈥� in Mochizuki鈥檚 proof. They argued that some of Mochizuki鈥檚 reasoning was flawed and that the ABC conjecture was still an open problem.

鈥淥pinion has definitely shifted toward the view that the proof is flawed since the letters from Scholze and Stix in 2018,鈥� says Andrew Booker at the University of Bristol, UK. 鈥淚t鈥檚 obviously bad for the [number theory] community if the result is declared a theorem in some circles but not others.鈥�

At a press conference on Friday in Kyoto announcing the paper鈥檚 acceptance, which Mochizuki did not appear at, mathematician Akio Tamagawa said the proof included no fundamental changes in response to Stix and Scholze鈥檚 criticism.

鈥淭he closest we鈥檝e come to this sort of dilemma in recent times is the controversy surrounding Thomas Hales鈥檚 proof of the Kepler conjecture in 1998,鈥� adds Booker. 鈥淚t was the other way around in that case, in that the proof was viewed as impenetrable but probably correct.鈥� Hales鈥� proof was eventually formally verified with the aid of a computer in 2014.

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A new record has been set for the largest encryption key ever cracked 鈥� but your secrets should be safe for now.

Long strings of numbers are essential to the encryption that keeps our online data safe. One widely used form of encryption called RSA cryptography relies on the fact that it is extremely difficult to find the prime numbers that multiply together to yield very large numbers.

The inventors of the RSA algorithm and challenged people to find the original primes, as a way of tracking how secure the encryption is against modern computers.

Now Emmanuel Thom茅 at the National Institute for Research in Computer Science and Automation in France and his colleagues have broken the record for the largest key cracked so far.

The team factored RSA-240, an RSA key that is 795 bits in size, with 240 decimal digits.聽The previous RSA record was set in 2010, with a key of 232 decimal digits and 768 bits.

鈥淲e were actually faster than the previous record, even though we computed something larger,鈥� says Thom茅.

The team also computed a discrete logarithm of the same size 鈥� these are essential for secure communications over computer networks, such as when a computer connects to a website securely using HTTPS.

Thom茅 and his colleagues ran computations across clusters of computers in France, Germany and the US. The total computing time took the equivalent of a single computer core running for 35 million hours, or almost 4000 years.

It took 8 million core hours to crack RSA-240, and computing the discrete logarithm was even more time-intensive, taking 27 million core hours.

The RSA keys most commonly used by ordinary computers today are larger in size, around 2048 bits, so the calculation isn鈥檛 a threat to computer security.

We would expect to crack larger and larger RSA keys as computing powers improves 鈥� a rule-of-thumb, known as Moore鈥檚 law, predicts that computing power doubles roughly every 18 months and can be used to determine when key sizes should be broken, given the time it took for previous records to be set, says Thom茅.

This time, the team managed to do it faster than expected, he says. 鈥淲e provided a new data point to make people able to determine how hard it should be now and, in the future, to compute things.鈥�

Read more: Theorem of everything 鈥� The secret that links numbers and shapes

2014 鈥� Maryam Mirzakhani (1977-2017)

2010 鈥� C茅dric Villani (1973- )

2006 鈥� Grigori Perelman (1966- ) [declined]

1998 鈥� Andrew Wiles聽 (1953- ) [silver plaque]

1990 鈥� Edward Witten (1951- )

1982 鈥� Alain Connes (1947- )

1966 鈥� Alexander Grothendieck (1928-2014)