In other words, Hilbert’s 10th problem is undecidable. Mathematicians hoped to follow the same approach to prove the extended, rings-of-integers version of the problem—but they hit a snag. Gumming Up the Works The useful correspondence between Turing machines and Diophantine equations falls apart when the equations are allowed to haveContinue Reading
The original version of this story appeared in Quanta Magazine. Most of life’s engines run on sunlight. Photons filter down through the atmosphere and are eagerly absorbed by light-powered organisms such as plants and algae. Through photosynthesis, the particles of light power a cellular reaction that manufactures chemical energy (inContinue Reading
The original version of this story appeared in Quanta Magazine. Calculus is a powerful mathematical tool. But for hundreds of years after its invention in the 17th century, it stood on a shaky foundation. Its core concepts were rooted in intuition and informal arguments, rather than precise, formal definitions. TwoContinue Reading
The original version of this story appeared in Quanta Magazine. Computer scientists often deal with abstract problems that are hard to comprehend, but an exciting new algorithm matters to anyone who owns books and at least one shelf. The algorithm addresses something called the library sorting problem (more formally, theContinue Reading
The original version of this story appeared in Quanta Magazine. Prochlorococcus bacteria are so small that you’d have to line up around a thousand of them to match the thickness of a human thumbnail. The ocean seethes with them: The microbes are likely the most abundant photosynthetic organism on theContinue Reading
The original version of this story appeared in Quanta Magazine. Pose a question to a Magic 8 Ball, and it’ll answer yes, no, or something annoyingly indecisive. We think of it as a kid’s toy, but theoretical computer scientists employ a similar tool. They often imagine they can consult hypotheticalContinue Reading
The original version of this story appeared in Quanta Magazine. Far from being solo operators, most single-celled microbes are in complex relationships. In the ocean, the soil, and your gut, they might battle and eat each other, exchange DNA, compete for nutrients, or feed on one another’s by-products. Sometimes theyContinue Reading
