Mathematicians put AI to work on Fermat's last theorem At an event in London, mathematicians have made unexpectedly fast progress on formalising Fermat's last theorem using AI In the lobby of a central London hotel, tourists are bracing themselves for a day of sightseeing in a heatwave. Meanwhile, staff are resetting the dining room after breakfast. And in a windowless meeting room, assembled academics are contemplating whether humans have a role to play in the future of mathematics, now that AI can prove theorems by itself. The general mood in the room is one of bewilderment at the recent jump in computer intelligence and excitement about the potential it unlocks - and perhaps a slight unease about what the future holds for them personally. Twenty-five researchers from diverse fields and countries are here to spend a week working on formalising Fermat's last theorem with cutting-edge AI models.

From AI with Hannah Fry to David Attenborough's early days, these are the five must-watch science documentaries of the year to date, says Bethan Ackerley In 2015, an amateur trophy hunter from the US shot and killed the largest lion in Africa. The vitriol unleashed after Cecil's death isn't surprising (or entirely unwarranted), but what is remarkable is how this delicately-crafted film uses the case as a locus for all sorts of arguments about conservation. A symbol in life and in death, Cecil and other large, charismatic animals exist in a complex balance with humans who, one way or another, invariably stake a claim on them. Almost everyone in the world now needs to have some knowledge of how AI technologies work, from all the chatbots they encounter to driverless cars and more. Mathematician Hannah Fry is an excellent person to impart such knowledge: across three episodes, she guides us through recent cases where AI has become entangled with very human problems.

Occam's razor has lost its edge. Can we sharpen our search for truth? Seeking out the simplest, most elegant explanations has served scientists well for centuries, but cognitive scientist Marina Dubova's experiments are revealing better ways to uncover reality Limited by the knowledge of his time, the ancient Greek astronomer Ptolemy imagined that the planets and sun of our solar system orbited Earth. Every new observation that pushed against this image required a slight tweak to that theory, until centuries later Nicolaus Copernicus's reimagining toppled it once and for all. A more elegant explanation proposed that all the planets orbited the sun, kicking off a scientific revolution that changed our understanding of the entire universe.

Peter Shor's algorithm could break the internet - but he's not worried Few people have invented an algorithm with the potential to spark a worldwide crisis, so why is quantum computing pioneer Peter Shor so unconcerned? "So, he's the Beyoncé of this event?" a young woman standing behind me says to a colleague. The three of us are standing, looking at the back of a crowd, whose members are all looking at a bearded man in an orange sweater. Getting a look at him is like trying to see the - only fleeting glimpses are possible. "His algorithm is the algorithm that will break everything," the colleague says, as I briefly catch sight of people posing for selfies and getting their conference badges signed.

Does time come from the entire universe running computations? Explaining the passage of time has been a gnarly problem in physics basically forever, but physicist and computer scientist Stephen Wolfram has a radical proposal for where it comes from. What if the universe is just one big computer? My colleagues and I have a running joke: time isn't real. Oh, you thought that deadline was tomorrow, but it's actually today?

Bees seem to show when they are pleased and like something, rather than just needing it, in one of the strongest signs yet that insects have subjective experiences. In recent decades, it has become clear that bees are capable of more complex behaviours than we previously thought, such as counting and demonstrating a sense of rhythm . But discerning whether they have inner states akin to our emotions is more difficult. For one thing, insects don't have the flexible facial musculature of mammals, which we use to communicate our feelings. "How can we get any behavioural readout of these insects with a hard body and their mask of a face," asks Andrew Barron at Macquarie University in Sydney, Australia.

How healthy is your brain? In our efforts to keep our brains healthy, how do we know what is working? It shouldn't have been difficult: 72 x 72. From the back seat, my daughter, newly confident in mental maths, wanted to check her answer. Whether it was because it was the end of the day, I was trying to park or something else, I stalled, cognitively speaking. Lately, though, I have had the sense that my brain isn't firing on all cylinders.

Can the biggest problems in AI be solved by philosophy? Some of the biggest challenges in artificial intelligence are being worked on not by computer scientists head down in code but by philosophers lured from academia into jobs at AI firms. The philosophers are tasked with making the next generation of models more capable and reliable, but they also shed light on the mystery of consciousness and whether intelligence can be replicated in software alone. Jonathan Birch at the London School of Economics and Political Science says AI companies are the big employers of philosophy PhDs right now, with offers of interesting work, large salaries and stock options proving too tempting for many to resist. "Topics that have been researched in philosophy departments for decades - how to make rational decisions, how to systematise moral principles, what counts as thinking or reasoning or introspection, what counts as evidence of consciousness - are suddenly of massive value to AI companies," says Birch. "So, naturally, we are seeing a huge brain drain."

From friendship in a world of chatbots to what it means to be alive, this month's new popular science books are asking some big questions. Australia's tiger quoll - as featured in Dan Werb's Our Wild Familiars, out this month Life, being alive and death are big themes in the new popular science books out in July, not to mention that small thing of being a human and all the messy feelings and sensory stuff that goes with it. Then there's also AI filling the future - in ways that worry one of the world's leading forensic scientists, as well as ethicists who are paid to think about this sort of thing. I'm looking forward to delving into the worlds of volcanoes and pharmacology, which look positively safe and stable in comparison Can friendship with a chatbot ever be as good as friendship with a gang of flesh-and-blood besties? Is there still and will there - can there - always be something about human friendships that will elude the smartest of simulations?

I am on holiday later this month, so I'm pleased to find there's a really wide range of intriguing new science fiction to take with me. I'm particularly keen to get cracking on a tale by Sheila Armstrong about strange ancient things found in a bog, but I'm also excited to read a new book by one of my favourite authors, Paul Tremblay (even if it does sound very disturbing). And I'm looking forward to the high-concept thrillers and classic space-set sci-fi on offer, too - not forgetting the first new novel released in 30 years. This sounds a little -like and ideal summer reading for those of us who enjoy a good high-concept thriller. It's set in a near future where you can outsource your emotional pain thanks to a biotech company, Eudaimonia.