This is a preview. Log in through your library . Abstract Carmichael's conjecture states that if φ(x) = n, then φ(y) = n for some y ≠ x (φ is Euler's totient function). We show that the conjecture is ...

For almost 200 years, Euler’s conjecture reigned in mathematics unchecked. Then, a paper came out that turned the whole thing on its head… and it did it all in just two sentences. In 1769, ...

Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses. “I’m nearly at the end of the talk, and it’s ...

Hannah Cairo found herself stuck on a problem that wouldn’t leave her mind. It wasn’t a regular homework assignment—it was a decades-old mathematical puzzle believed to be true by leading experts in ...