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Mathematics
number theory
A Numerical Mystery From the 19th Century Finally Gets Solved
Two mathematicians have proven Patterson’s conjecture, which was designed to explain a strange pattern in sums involving prime numbers.
number theory
Mathematicians Crack a Simple but Stubborn Class of Equations
Ever since Archimedes, mathematicians have been fascinated by equations that involve a difference between squares. Now two mathematicians have proven how often these equations have solutions, concluding a decades-old quest.
mathematical physics
At Long Last, Mathematical Proof That Black Holes Are Stable
The solutions to Einstein’s equations that describe a spinning black hole won’t blow up, even when poked or prodded.
Insights puzzle
Seeking Mathematical Truth in Counterfeit Coin Puzzles
Readers balanced logical reasoning and mathematical insights to find phony coins with a double-pan balance scale.
geometry
A Question About a Rotating Line Helps Reveal What Makes Real Numbers Special
The Kakeya conjecture predicts how much room you need to point a line in every direction. In one number system after another — with one important exception — mathematicians have been proving it true.
Quantized Academy
How Can Infinitely Many Primes Be Infinitely Far Apart?
Mathematicians have been studying the distribution of prime numbers for thousands of years. Recent results about a curious kind of prime offer a new take on how spread out they can be.
combinatorics
Hypergraphs Reveal Solution to 50-Year-Old Problem
In 1973, Paul Erdős asked if it was possible to assemble sets of “triples” — three points on a graph — so that they abide by two seemingly incompatible rules. A new proof shows it can always be done.
The Joy of Why
How Do Mathematicians Know Their Proofs Are Correct?
What makes a proof stronger than a guess? What does evidence look like in the realm of mathematical abstraction? Hear the mathematician Melanie Matchett Wood explain how probability helps to guide number theorists toward certainty.
general relativity
Mass and Angular Momentum, Left Ambiguous by Einstein, Get Defined
Surprising as it may sound, 107 years after the introduction of general relativity, the meanings of basic concepts are still being worked out.