OpenAI has revealed solutions to a number of long-standing mathematics problems produced by an unreleased frontier model in a batch of 722 manuscripts, covering 372 result families that group related papers. It extends a run of breakthroughs that have both impressed and unsettled parts of the mathematical community while raising questions about research ethics and academic conduct. According to AGMAI, the newly formed independent advisory group of elite mathematicians put together to work on communicating the results responsibly, the release includes solutions to "hundreds" of open questions.The results have been expected for weeks, althou …Read the full story at The Verge.

I have close to 10TB of data on my NAS, including backups and a large movie library for Jellyfin. Some of that storage is used to keep multiple copies of the same files, and I’m responsible for a fair bit of it. I sometimes download a movie in 1080p, add the 4K release when it becomes available, and leave the older version sitting there. I also had repeated backups and plenty of identical videos saved under different names piling up.

The Dreame X60 Ultra in action. | Photo by Jennifer Pattison Tuohy / The Verge Spending too much time constantly sweeping and mopping? If you have the money to delegate those chores to a robot cleaner, you could save a lot of personal time. Thankfully, it’s cheaper than usual to buy a robot vacuum / mop hybrid during Amazon’s Prime Big Deal Days, and some of our favorite models are discounted. The deals aren’t just for Amazon Prime members either, with other retailers like Best Buy and manufacturers getting in on the action, as well. That being said, if your floors are already squeaky clean, make sure to check out our roundup of over 100 deals from Amazon’s October sale.Roborock Qrevo Curv 2 FlowRoborock’s first roller-mop robot vacuum, the Curv 2 Flow, features flagship specs at a midrange price, including excellent mopping, a retractable mop cover, 20,000Pa suction, AI-powered obstacle avoidance, and a compact design.Where to Buy: $999.99 $634.99 at Amazon $999.99 $699.99 at Best Buy $999.99 $699.99 at RoborockRoborock Saros 20Roborock’s Saros 20 stands out for its ability to climb tall thresholds, avoid obstacles with impressive accuracy, and clean under low furniture thanks to its slim design. It also combines powerful 36,000Pa suction with great spinning mop pads, which is why we recommend it for homes with multiple floor types and tricky transitions between rooms.Where to Buy: $1599.99 $1179.99 at Amazon $1599.99 $1194.99 at RoborockNarwal Flow 2Our pick for the best robot vacuum-mop if you have all hardwood floors is the Narwal Flow 2. A superb mop with a wide flat roller pad and an excellent vacuum, the Flow 2 also has top-notch navigation and obstacle detection. Its sleek base station is also less of an eyesore than most.Where to Buy: $1499.99 $1099.99 at Amazon (with on-page coupon) $1499.99 $1044.99 at NarwalDreame X60 UltraThe Dreame X60 Ultra leverages a pair of AI-assisted cameras to help it navigate obstacles and ensure a thorough, consistent cleaning. It’s only 3.13 inches tall, so it can sneak under low surfaces that some other robot vacuums can’t.Where to Buy: $1499.99 $1099.99 at Amazon $1499.99 $1199.99 at Best Buy $1499.99 $1114.99 at Dreame (with code RVFD3)Eufy Omni C28The Eufy Omni C28 combines excellent mopping performance with a small self-cleaning dock that’s well suited for apartments and other smaller spaces. It navigates around furniture and into tight areas with ease, while its roller mop does a great job cleaning hard floors.Where to Buy: $799.99 $449.99 at Amazon $799.99 $449.99 at Eufy (with code WS7DV2MUCMGY)Dreame L40 UltraDreame’s budget-friendly L40s Ultra has storage for up to 100 days of dust, and can clean its own mop pads. The auxiliary climbing wheels help it navigate thresholds and rug edges with ease.Where to Buy: $799.99 $399.99 at Amazon $799.99 at DreameRoborock Q10 S5 PlusA budget robot vacuum and mop without a multifunction dock. It has AI-powered obstacle avoidance, sonic mopping, auto mop lifting, and an auto-empty dock.Where to Buy: $549.99 $249.99 at Amazon $549.99 $264.99 at Wellbots (with code NEW10) $549.99 $264.99 at RoborockBest overall: Deebot X8 Pro OmniThe Deebot X8 Pro Omni is an excellent roller mopping bot that extends to clean along backboards and furniture. It’s also a powerful vacuum with great obstacle avoidance, a small dock, and a convenient self-cleaning brush system.Where to Buy: $1399.99 $699.99 at Best BuySwitchbot K11 PlusThe Matter-compatible SwitchBot K11 Plus is ideal for small apartments, bedrooms, and home offices. It can easily navigate around chair and desk legs and into tight spaces that most other bots miss. It also features a commendable 6,000Pa of suction power — which is great for this price point — and works well on both hardwood and carpet.Where to Buy: $399.99 $199.49 at Amazon (black) $399.99 $200 at SwitchBot (with code FSALE50) $399.99 $199.49 at Amazon (white)

I love my Nvidia Shield TV Pro, but I recently decided it was time to replace my seven-year-old streaming box. The only thing currently on the market that comes close to its performance is the Onn 4K Android TV budget box. It has newer hardware than the Google TV Streamer, which is still too buggy. Not to mention, the price was quietly increased by $50.

Whenever Claude Code finishes a build, it will let you know, and the file usually opens and will mostly work. What it doesn't tell you is what's off with the first look, and if you don't actually write code, like me, then there isn't much to check it against. Reading line by line isn't realistic for me so I have to rely on testing the build and asking the agent what's happening underneath.

Google announced a new agreement to update six nuclear power plant sites across the US as the tech giant seeks to generate more electricity for its power-hungry data centers.Google signed the 20-year deal with Constellation, the leading nuclear power plant operator in the US. The power purchase agreement is meant to guarantee the revenue needed to cover the cost of efficiency upgrades at existing nuclear plants and "represents more than $4.3 billion of new investment by Constellation," according to the press release.Over the next five years, the companies expect new equipment at each site to collectively lead to 890 megawatts of addition …Read the full story at The Verge.

Comedian Sebastian Maniscalco is facing backlash from fellow comics who claim his takeover of SiriusXM's Raw Comedy channel is harming up-and-coming talent, as reported earlier by Deadline. Many established comics, including Patton Oswalt and Margaret Cho, have posted videos online calling on Maniscalco to carry on Raw Comedy's legacy by highlighting newer comedians on his channel, instead of mostly focusing on well-known ones.SiriusXM launched Sebastian Maniscalco's Comedy Radio in April, which is "curated and headlined by Sebastian himself," according to the satellite radio provider. The move replaced SiriusXM's long-running Raw Comedy c …Read the full story at The Verge.

I’ve used Microsoft 365 for years, so moving away from it felt like a bigger experiment than simply uninstalling a few apps. I wanted to see how far I could go with free, open-source software without making my daily workflow harder. Instead of looking for one package to replace everything, I tested different tools for different jobs and used them long enough to see where they actually fit. Everything worked more smoothly than I expected, but one part of the experience made me realize just how much I had taken a certain Microsoft app for granted.

Apple is collaborating with LG on a new lineup of smart home devices, including a video doorbell, thermostat, indoor camera, and more, according to a report from Bloomberg. The products will reportedly feature LG branding but integrate with the company's rumored smart home hub that's expected to launch during an event on October 13th.Sources tell Bloomberg that the upcoming line of devices also includes a deadbolt lock, outdoor security camera, and a floodlight camera. Several of the devices will reportedly come with support for the smart home protocols Thread and Matter amid a "formal reboot" for Apple's HomeKit platform. T …Read the full story at The Verge.

Xbox CEO Asha Sharma told employees that Microsoft is getting ready to do something around Grand Theft Auto VI that "no other platform holder is doing" during an employee all-hands this morning. According to sources familiar with Microsoft's plans, Xbox has managed to land a deal for the exclusive game streaming rights for Grand Theft Auto VI.The deal will mean only Xbox Cloud Gaming can stream Grand Theft Auto VI to players at launch, but it's not clear how long the agreement will be in place. Rockstar Games still hasn't announced a PC version of Grand Theft Auto VI, so this deal will allow Xbox to market its Xbox Cloud Gaming service to …Read the full story at The Verge.

Linux runs nearly everything I need now, and for that reason, it's my daily driver OS. There are exceptions, and they're almost exclusively games that require an anti-cheat client that doesn't play well with Linux. These are games like EA Sports FC 27, Escape From Tarkov, Battlefield 6, and most other competitive games. Those games also happen to be some of my most played titles, which is why I've kept around a Windows install on a second drive for the better part of a couple of years.

Some Tesla Model 3 and Model Y owners can use their EV battery to keep the lights on longer during a power outage, now that Tesla's expanding the Powershare Home Backup feature. It was previously only available on the Cybertruck, and also requires owning Tesla's Powerwall 3. However, support for the Powerwall 2 and Powerwall Plus is "coming soon." Tesla's EVs aren't the first or only ones with bidirectional charging; companies like Kia and GM also offer vehicle-to-home power on their vehicles, while Nissan's Leaf offered power-to-grid charging years ago.Tesla says the Model 3 and Model Y can extend home backup power to over two days for Pr …Read the full story at The Verge.