A mathematician has somewhat unceremoniously announced that he has disproven an almost 90-year-old problem — with the help of Anthropic’s artificial intelligence model. The announcement sent waves through social media and mathematics circles, as AI-driven mathematical achievements have shaken the field and called the future of math into question.

What math problem did AI tackle?
The Jacobian conjecture is “false,” Harvard University mathematician and Anthropic employee Levent Alpoge said on X. Attached was a mathematical formula, a counterexample to the decades-old problem. The algebraic geometry theory suggests that a “certain type of mathematical function would also work in reverse,” said New Scientist. Despite the fact that the problem “bedeviled highly accomplished mathematicians for almost 90 years,” the conjecture “proved no obstacle” for Anthropic's “latest frontier AI model,” said Mashable.

The AI-generated counterexample that Alpoge shared online is the “latest in a series of AI-driven mathematical breakthroughs,” said Fortune. The technology’s “progress (or assault)” on unsolved mathematics has “compounded rapidly since mid-2025,” when models first solved “five of six problems” at the International Mathematical Olympiad. Mathematicians, “relegated to the role of shepherd,” are left to watch AI “close these questions one-by-one, reaching places where the human mind can’t follow.” The mathematicians’ reaction is a “now familiar mixture of dread and amazement.”

What does this mean for the future of math?
The significance of the AI’s discovery varies depending on whom you ask. The counterexample did not “cause me to update my priors about what AI can and can’t do," professor and mathematician Andrew Blumberg said to Mashable. This is “exactly the kind of thing I would expect AI to be able to do.”

For some, the recent developments of AI foreshadow a future of “increasingly capable AI models that will solve ever more complex problems,” said New Scientist. AI can “already produce master’s degrees in mathematics,” mathematician Ivan Fesenko at Westlake University in China said to the outlet. In a year, it will produce “Ph.D. degrees in mathematics.” Then the question arises: “Do we really need so many mathematicians around if AI can do such things so nicely?”

This is the summer of “cerebral pursuits” in France, with a “big rise” in the sale of cahiers de vacances, an adult spin-off of school vacation workbooks, said The Times. It’s the grown-ups that are “brushing up” on algebra, history and grammar on the beach.

All grown up
Vacation workbooks were originally designed to help schoolchildren keep up with their studies over the long summer break. But the new adult versions are a real hit. You only have to scan bookshop shelves to see the “scale of the phenomenon,” said the nonprofit Aleteia. French adults are using the workbooks not only to “evoke childhood summers” but also as an entertaining way to learn something new during the summer holidays.

Last year, more than 650,000 vacation workbooks were sold — double the year before — and publishers are reporting even bigger sales this summer, said The Times. Among the most popular ones are “Les Maths en Vacances,” which has readers practicing geometry by folding their beach towels, and “Énigmes de l’Histoire,” which focuses on historical puzzles, such as the “mysterious fate of Louis XVII,” who disappeared after his parents were beheaded during the French Revolution. “Passeport Adultes Anglais” is all about improving your English, with a “notably challenging” section on making British “herb jelly.”

There are also logical reasoning challenges. And police investigation workbooks offer “puzzles to solve” in a “Clue”-style format, said Instant.

‘Need to disconnect’
Adults want to buy these books because they allow them to “learn a lot while having fun,” said Didier Ferat, of workbook publisher Solar, to Aleteia. The format is not “intimidating,” said Carole Strang, of publisher Éditions Nathan, to Instant. Readers can work through the books “alone or with others.” 

There’s a sense of nostalgia at play too. Everyone knows these workbooks from their school days. They are associated with “strong memories” of childhood summers and holidays, said Strang.

The desire to disconnect through nostalgic activities also explains the associated boom in “kidult” coloring and sticker books and an increasing passion for puzzles and crosswords. Dictionary publisher Robert has launched a new series of themed giant crossword books this summer. At a time when the “world of books is suffering,” said the publisher Charles Bimbenet to Instant, “this enthusiasm is welcome.”

Red-crowned and white-naped cranes play in China’s Zhalong Nature Reserve. The area’s grasslands have become a major breeding ground for rare cranes in northeastern China. 
Chi Shiyong / VCG / Getty Images

Phenomenally delicious plant-based eating has long existed in many of the world’s great cuisines. As with any kind of cooking, how a dish tastes is all in how it is made. These books will show you how to achieve top-tier vegan dishes from different angles.

‘The Korean Vegan Cookbook: Reflections and Recipes from Omma’s Kitchen’ 
You may already know Joanne Lee Molinaro, aka The Korean Vegan, from social media. In her debut book, Molinaro “masterfully intertwines delicious recipes with intimate moments from her life,” said Bon Appétit. Americans might think of Korean food as meat-centric, but “Korean temple food, the cuisine prepared by Korean buddhist monks, has been around for a thousand years, predating Korean BBQ.” ($37, Avery)

‘Superiority Burger Cookbook: The Vegetarian Hamburger Is Now Delicious’ 
Come for Brooks Headley’s recipe for the New York City restaurant’s legendary vegetable burger, stay for the delicious dishes that orbit that burger’s well-deserved fame. Making the burger is an ordeal of sorts, but the result is a paragon of the genre: “crisp outside and a soft but not-boring inside that’s lightly spiced and spicy, with plenty of vegetables and meat-free umami,” said Edible Manhattan. ($30, Norton)

‘Vegetable Kingdom: The Abundant World of Vegan Recipes’ 
In “Vegetable Kingdom,” food justice advocate Bryant Terry skips across the Black and African diaspora. Spinach salad with blackened chickpeas, a panko-crusted cauliflower take on the Kenyan chicken, jerk tofu enveloped in collards — this is a book for “anyone who wants to make their beans and greens more exciting,” said Bon Appétit. ($30, Ten Speed Press)