One of the most bitterly contested proofs in modern mathematics may be on the verge of being untangled. Two projects, both aiming to use a computer program to cast new light on the controversy, are now up and running - with one having operated in secret for more than two years already. The developments are a positive sign that the row might find a solution, say mathematicians. The saga began in 2012 when Shinichi Mochizuki at Kyoto University, Japan, claimed to have proved a famous idea called the ABC conjecture, posting a 500-page proof online. The conjecture is simple to state, concerning prime numbers involved in solutions to the equation a + b = c and how these numbers relate to each other.

I don't see images in my head. Can training give me a mind's eye? Training programmes for people with aphantasia - the inability to create mental images - are challenging neuroscientists' understanding of how we create thoughts What do you see when you try to picture an apple? Last December, I closed my eyes and tried to visualise a potoo. This tropical bird has a "round, kind of pill-shaped head", my mental imagery coach described to me, and is covered with brown feathers. Its cartoonishly large mouth opens like a gaping smile to reveal a pink, fleshy colour, and its large irises can make its eyes seem entirely black.

Why the world's militaries are scrambling to create their own Starlink The reliable internet connections provided by Starlink offer a huge advantage on the battlefield. Starlink's satellite constellation provides a reliable internet connection to almost anywhere on Earth, conferring an advantage on the modern battlefield. But it is also run by controversial billionaire Elon Musk, presenting a risk to militaries that could easily find themselves cut off. So, now countries are racing to build their own version. The Starlink network consists of almost 10,000 satellites that offer internet connections across most of the planet via small dishes on the ground.

The US and Iran are trading blows in the Gulf with a simple drone that costs as little as $50,000 to make. But why is a slow, cheap and relatively primitive drone seeing use in 2026 alongside hypersonic missiles and stealth jets? Iran invented the relatively simple Shahed 136 attack drone, but is now fending off US copies launched against it in combat. Why, when the US military has expensive, cutting-edge and hi-tech weapons, is it making flimsy drones powered by a motorbike engine? Iranian company Shahed Aviation Industries originally designed the 136.

Oshen's robotic sailboats are powered by the wind and the sun The UK's Royal Navy may return to the age of sail, with a new demonstration involving a flotilla of small, wind-propelled robot boats. Made by Oshen in Plymouth, UK, the vessels, known as C-Stars, are just 1.2 metres long and weigh around 40 kilos. Solar panels power navigation, communications and sensors, while a sail provides propulsion. Deployed as a constellation, the small vessels act as a wide-area sensor network. How the US military wants to use the world's largest aircraft "The simplest way of describing C-Stars is as self-deploying, station-keeping ocean buoys," says Oshen CEO Anahita Laverack .

The ideas presented in George Lakoff and Srini Narayanan's The Neural Mind are fascinating, but the writing is far less compelling This is a book review in two parts. The first is about the ideas presented in The Neural Mind: How brains think, which are fascinating. The second is about the actual experience of reading it. The book tackles one of the biggest questions in neuroscience: how do neurons perform all the different kinds of human thought possible, from planning motor actions to composing sentences and musing about philosophy? The authors have very different perspectives.