A Chinese artificial intelligence system has solved a decade-old problem proposed by an American mathematician, according to a new study.

The algebra conjecture was first posed in 2014 by then University of Iowa professor Dan Anderson, who died in 2022.

An AI system developed by a Peking University team processed decades of mathematical literature to crack Anderson’s problem and verify its own findings without any human intervention.

“Using this framework,” the team said in a yet-to-be peer-reviewed study posted in the arXiv repository, “we successfully solved an open problem in commutative algebra and automatically formalised the proof with essentially no human intervention.”

Scientists observed that the AI system could perform mathematical tasks faster than any human, including independently doing work that normally required collaboration between different field experts.

“This work provides a concrete example of how mathematical research can be substantially automated using AI,” the researchers, led by Peking University mathematician Dong Bin, said.

AI systems are being trained across the world to solve mathematical problems, but they still require a large amount of human supervision to crack math problems. “Mathematical proofs demand complete rigour, yet even expert-written proofs may contain subtle flaws and proofs produced by LLMs, which are prone to hallucination, are far less reliable,” the Chinese scientists wrote in the latest study.

“Motivated by this, we propose a framework for autonomously tackling and verifying research-level mathematics that integrates a natural language reasoning agent with a formalisation agent.”

The new AI applies a reasoning system called Rethlas, which draws from the maths theorem search engine, or Matlas, to explore strategies for solving a problem, following a workflow similar to what mathematicians use.

When Rethlas comes up with a potential proof, a second system called Archon uses another search engine called LeanSearch to transform the proof into a project for an interactive theorem prover.

This theorem prover, Lean 4, is also a programming language with a community-maintained library that has hundreds of thousands of theorems and definitions.

Researchers used the new AI system to solve Anderson’s algebra conjecture within 80 hours of runtime.

“No mathematical judgment was required from the human operator,” they wrote.

However, researchers found that they could speed up the process if a mathematician guided Archon.

“Our work illustrates a promising paradigm for mathematical research in which informal and formal reasoning systems, equipped with theorem retrieval tools, operate in tandem to produce verifiable results, and substantially reduce human effort,” they noted.

A 71-year-old man died 10 days after the fight in March 2025

A man who was involved in a fight where a elderly man died has pleaded guilty to manslaughter. Suffolk Police was called to a fight involving four people in Newmarket High Street at around 3.40pm on March 8, 2025.

Two men sustained injuries and were taken to hospital. One of the men, aged 45, was discharged the same evening. The other man involved, a 71-year-old, remained at Addenbrooke’s Hospital in a critical condition. However, he later died on March 18.

Levi Williams, of Holland Park, Newmarket, was arrested at the scene on suspicion of assault causing grievous bodily harm. He was initially bailed before later being rearrested and charged with murder.

Appearing at Peterborough Crown Court on Friday (April 10), he pleaded guilty to manslaughter. He’s been bailed and is due to be sentenced on June 4 at the same court.

A 24-year-old man, who was arrested on suspicion of assault causing grievous bodily harm, and a 45-year-old man, who was arrested on suspicion of affray, have been released under investigation while enquiries continue.

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Officers are appealing for information about the incident in Nesfield Close, Harrogate, shortly before 2.30am on Monday, April 6.

North Yorkshire Police said the criminal damage happened after a person approached Nesfield Close from the direction of Stonebeck Avenue.

The force has urged anyone with information to get in touch.

“Anybody who may have any information, CCTV or doorbell footage please email sarah.danby@northyorkshire.police.uk,” a force spokesperson said.

“Alternatively you can call North Yorkshire Police on 101 or contact Crimestoppers anonymously on 0800 555 111 or via their website.

“Please quote reference 12260060906 when referring to this incident.”

A passenger on the service between Bolton and Leyland became ill while riding the train.

Northern Rail posted a message to their X account at 1:50pm alerting people that this was the case.

Some Northern lines were blocked or delayed as a result of the medical incident.

A Northern Rail spokesperson was not able to provide much information on what occurred, but confirmed that the issue was over around 20 minutes after it began.

The blockages were quickly cleared and Northern Rail service returned to normal.

Northern Rail posted another X update at 2:09pm alerting passengers that the service was back to normal.

Riders may be entitled to compensation if their train was cancelled or delayed by longer than 15 minutes.

Police would like to speak to two men captured on CCTV

A man was left injured after an assault in a Cambridgeshire city. The victim was assaulted in Market Street, Ely at around 2.25am on March 29.

The man suffered an injury to his arm. Cambridgeshire Police would like to speak to two men captured on CCTV in connection with the incident.

A police spokesperson said: “Anyone with information should call police on 101 quoting 35/23024/26 or visit www.cambs.police.uk/report. Alternatively, contact Crimestoppers, anonymously, on 0800 555111 or via www.crimestoppers-uk.org.”

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“Gary is very much sick at home, he’s under the care of the NHS and he’s not doing well, but he is focused on rest and recovery,” she said.

Newly co-opted DUP MLA Julie Middleton has said her husband is “not doing well” as she steps into his role at Stormont, describing an “incredibly difficult time” for their family.

The new MLA for Foyle was speaking at a press conference on Monday morning, hours after being formally co-opted to replace Gary Middleton following his decision to resign on mental health grounds.

Setting out her priorities, she sought to strike a balance between acknowledging the personal circumstances behind her appointment and asserting her independence as a political figure in her own right.

“I am hugely honoured and excited to be here today as the first female unionist MLA for Foyle. I have a history and experience of working within the council as an elected representative for the DUP, and I bring forward a strong mandate from there and a strong record of delivery. I was born and bred in the Foyle constituency.

“I think it’s arguably one of the best constituencies, but I’m not naive to the issues that exist within the constituency in terms of deprivation, a need for further investment and indeed very close to home, but mental health issues as well,” she said.

“I have a history of working with young people, some of the most vulnerable young people in the Foyle constituency, for many years, and I also have a history of working within the special educational needs sector. I have a positive outlook on life.

“Things have been difficult at home. But I’m not just a wife, I am very much my own person, and I think that through this whole journey our family has been through, it’s clear that there needs to be further talks about mental health. There is certainly a stigma that still exists, but onto the work ahead, I’m very excited to be here today.

“I’m really looking forward to being in the chamber, and I’m looking forward to meeting with constituents on the doors and out and about, where they can tell me the things that are problems for them, how I can work for and with them and the things they want to celebrate and how I can be a cheerleader for them and the area as well. I am taking this very seriously, and I’m very, very pleased to be here today.”

Addressing questions about her husband’s condition, Mrs Middleton was direct about the seriousness of his illness while defending her decision to take on the role.

“It’s been an incredibly difficult time for our family, and I certainly have learned so much about mental health, and I do think the general public needs to understand that mental health is more than just your situation.

“Mental health is an illness within your body and certainly over the last 48 hours I have seen a small amount of comments regarding how could I possibly step into this role if Gary is sick at home.

“Well, Gary is very much sick at home, he’s under the care of the NHS and he’s not doing well, but he is focused on rest and recovery. I think what most people who don’t know me, maybe they’re not aware of the fact that I have been in the public eye throughout this whole journey and our private life and our public life are two different things. So Gary is unwell. He needs to step back to focus on rest and recovery.

“I am very much my own person. I am very excited to represent the area, but importantly, I am going to bring my own energy, my own enthusiasm. I’m known to be a joyful person, I’m known to be an open person, and I’m known to be a woman of integrity in the area that I’ve represented.

“I would say that everybody that I’ve represented will tell you I haven’t ever approached any work at a 50 per cent rate. I’m 100 to 110%. I put my whole heart and soul in there, and simply because I want to make people’s lives better, and words are just words, but words are stronger and more important when you back them up with action, and that is what I’m about.”

She also suggested her husband’s illness should prompt a wider conversation about mental health support, while cautioning against attributing blame to political life itself.

“I believe there needs to be carefully planned mental health interventions in every industry. I have already said that the illness is not caused by the work, and Gary hasn’t said that either.

“Gary hasn’t shared his story yet, but when he does share his story, it is a powerful one. He is stepping away from the public eye to rest and recover, not because Stormont or politics have caused his illness.

“He is ill with mental health, but we have a stigma in Northern Ireland, not amongst everyone, but amongst a minority of people, where they may not understand mental health as it is medically and I think that every workplace needs to focus on mental health.

“Every home needs to focus on mental health and it’s just a bigger piece of work than just talking things through right now, but I know that there are many that are engaged actively and I know that there are many who are really passionate about it. There are great organisations out there, and I am hugely thankful to the NHS for the help that they’ve given my family.”

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Pope Leo, the first U.S.-born pontiff, has fired back after President Donald Trump lashed out at him over the ongoing Iran war.

Trump derided Leo as “terrible” in an unusual, direct broadside against the leader of the Catholic church, who responded by vowing to continue denouncing war and suffering.

“I have no fear of the Trump administration, or speaking out loudly of the message of the gospel, which is what I believe I am here to do, what the Church is here to do,” he told reporters Monday. “I don’t want to get into a debate with [Trump].”

“I will continue to speak out loudly against war, looking to promote peace, promoting dialogue and multilateral relationships among the states to look for just solutions to problems,” Leo continued aboard a papal flight to Algiers, where he is embarking on a 10-day tour to four African countries.

“Too many people are suffering in the world today,” he said. “Too many innocent people are being killed. And I think someone has to stand up and say there’s a better way.”

Archbishop Paul S. Coakley, president of the U.S. Conference of Catholic Bishops, said he was also disheartened by Trump’s comments.

“Pope Leo is not his rival; nor is the Pope a politician. He is the Vicar of Christ who speaks from the truth of the Gospel and for the care of souls,” he said in a statement.

Trump is facing widespread backlash for his comments, and for sharing an AI picture of himself as Jesus.

“Pope Leo is WEAK on Crime, and terrible for Foreign Policy,” Trump wrote in a stunning 334-word Truth Social post on Sunday night while flying back to Washington, D.C. from Florida.

“I don’t want a Pope who thinks it’s OK for Iran to have a Nuclear Weapon,” he continued. “I don’t want a Pope who thinks it’s terrible that America attacked Venezuela, a Country that was sending massive amounts of Drugs into the United States.”

He claimed Leo — formerly Cardinal Robert Francis Prevost — was elected pope only because of the president, and slammed him for meeting last week with David Axelrod, who served as a senior advisor to former President Barack Obama.

“Leo should get his act together as Pope, use Common Sense, stop catering to the Radical Left, and focus on being a Great Pope, not a Politician,” the president concluded.

Less than an hour later, Trump posted an AI image portraying himself as Jesus appearing to heal a sick man, encircled by bald eagles and fighter jets.

He continued his tirade against the pontiff when his plane touched down, telling reporters on the tarmac: “I’m not a fan of Pope Leo.”

His comments came after Leo suggested over the weekend that a “delusion of omnipotence” triggered the Iran war. Late last month, he wrote on X that God “does not listen to the prayers of those who wage war, but rejects them.”

And, before the ceasefire, when Trump threatened to strike Iranian infrastructure and warned that “an entire civilization will die tonight,” Leo described these remarks as “truly unacceptable.”

The billionaire president’s scathing attack on the Catholic Church’s leader drew broad backlash in the U.S. — including from some members of his own party.

“The deranged and disgusting post from Trump attacking Pope Leo should certainly help him appeal to the more than 50 million Americans who identify as Catholics,” Rep. Pramila Jayapal, a Washington Democrat, wrote on X.

“Trump shamefully attacked His Holiness Pope Leo XIV,” House Minority Leader Hakeem Jeffries said in a statement. “People of faith will never worship a wannabe King. We worship an almighty GOD.”

“On Orthodox Easter, President Trump attacked the Pope because the Pope is rightly against Trump’s war in Iran and then he posted this picture of himself as if he is replacing Jesus,” former Georgia GOP Rep. Marjorie Taylor Greene wrote on X. “I completely denounce this and I’m praying against it!!!”

The Iran war, which has engulfed large swaths of the Middle East in violence, leading to the deaths of thousands, is opposed by a majority of Americans, according to multiple recent polls.

Earlier this month, it was reported that the American Pope may not return to the U.S. while Trump remains in office.

The Free Press reported that in the wake of the Pope’s annual address to the Vatican’s diplomatic corps in January, in which he criticized states seeking conflict around the globe, the Department of Defense invited Cardinal Christophe Pierre for a meeting.

At the time, Pierre was serving as Pope Leo’s personal envoy to the United States. The Pentagon requesting a meeting with a Vatican official is “unprecedented,” according to the report.

According to Vatican and U.S. officials who were briefed on the meeting and spoke to the outlet, the Pentagon criticized the Pope’s statements in January, interpreting them as hostile attacks on Trump’s policies. U.S. officials reportedly lectured the church representatives, with one even invoking the the Avignon Papacy —a period of time when the French government used military force to exert influence over the Vatican.

A Pentagon spokesperson told Newsweek that The Free Press’ reporting was “highly exaggerated and distorted.”

“I would actually like to talk to Cardinal Cristophe Pierre and, frankly, to our people, to figure out what actually happened,” Vice President JD Vance, a Catholic, told reporters last week. “I think it’s always a bad idea to offer an opinion on stories that are unconfirmed and uncorroborated, so I’m not going to do that.”

The case will be covered across two episodes

A famous Cambridgeshire cold case is set to feature across two episodes of a TV show on Channel 4 starting tonight (Monday, April 13). 24 Hours in Police Custody is returning to Channel 4 with the case of Una Crown, 86, who died at her home in Wisbech and here is how you can watch it.

The widow was found dead by her nephew-in-law on January 13, 2013, in a pool of blood with her body being burned. Two police officers attended to the scene and concluded her death was not suspicious, believing she had accidentally set herself on fire and tried to put it out with a tea towel.

The investigation concluded the two cuts to her throat were likely to have been caused by pressure from her scarf as she fell to the floor. A later post-mortem determined Crown had been stabbed to death and a murder investigation was launched.

However, Crown’s house had already been cleaned, as it was not originally treated as a murder investigation crime scene. The investigation eventually went cold for a decade before being reopened after a DNA breakthrough in the case.

Channel 4’s popular series, 24 Hours in Police Custody be will airing a two-part special called The Cold Case Murder which follows Detective Superintendent Iain Moor and colleagues in the Major Crime Unit, as they look into the new evidence to catch her killer.

The episodes will look at how the team used new forensic techniques to match DNA found under Crown’s fingernails to a man who lived just down the road named David Newton. You will see how detectives had to travel all over the county to rule out his male relatives and create a strong prosecution case.

Newton, who is now in his 70s, was sentenced to life in prison with a minimum of 21 years in February last year thanks to the team’s investigation. Judge Justice Neil Garnham described Newton as having carried out a “ferocious and sustained knife attack on a defenceless old lady”.

If you are interested in watching 24 Hours in Police Custody: The Cold Case Murder to find out more about how Det Supt Moor and his team were able to finally solve Crown’s murder, the first episode will be airing tonight on Channel 4 at 9pm. The second episode will on the same channel at 9pm tomorrow (Tuesday, April 14).

If you miss the episodes, you can watch them after they have aired on Channel 4’s streaming service.

“Bottlers!”

It’s the cruel taunt thrown about by rival fans. But you’ve gotta be in the Premier League title race to win it, right?

It’s just when you lead for so long and don’t win it, there’s a lonely void, an ‘it should have been us’ vacuum of sorrow.

Staring at the prospect of letting another seemingly insurmountable lead slip, Arsenal are desperate to banish those demons, having grappled with the tag over the past two decades.

But they are far from alone – in fact, the Gunners’ slip-ups have been far less dramatic than those of some of their rivals.

To help investigate the top flight’s biggest wobblers, those who have let the league slide from seemingly watertight positions, we enlisted the help of BBC Sport statistician Chris Collinson, someone always on top of his tables.

While Chris’ research shows us Arsenal deserve to feature several times, there is room for Stevie G’s slip, a mention for that ‘Aguerooo!’ moment, and, of course, Kevin Keegan’s iconic monologue – which took place 30 years ago this month.

And I’ll tell you, honestly, I will love it if you read this, love it!

Fatou, the world’s oldest gorilla living in captivity, has marked her 69th birthday with a special, sugar-free feast at Berlin Zoo.

The primate enjoyed a spread of cherry tomatoes, beets, leeks, and lettuce, eschewing traditional cake for a healthier, age-appropriate meal.

A western lowland gorilla, Fatou arrived in what was then West Berlin in 1959, believed to be around two years old at the time.

While her exact birthdate remains unknown, April 13 is celebrated as her designated birthday. Gorillas typically live for 35-40 years in the wild, but can achieve significantly longer lifespans under human care.

Fatou became the zoo’s most senior resident in 2024 following the passing of Ingo the flamingo, who was thought to be at least 75 and had resided at the zoo since 1955.

Fatou’s remarkable journey began in the wild of western Africa, with Guinness World Records noting a tale that a French sailor bartered her to settle a bar tab in Marseille, before she was reportedly sold to the zoo by a French animal trader.

Today, Fatou enjoys her own enclosure, preferring a quiet life away from the zoo’s younger gorillas.

She has lost her teeth and experiences some arthritis and hearing loss. Despite these age-related challenges, Christian Aust, the Berlin Zoo’s primate supervisor, describes her as friendly with her keepers, albeit still a little stubborn.

At 69, it seems she has certainly earned the right.