## Latest Articles

### From Systems in Motion, Infinite Patterns Appear

Mathematicians are finding inevitable structures in sufficiently large sets of integers.

### How Cryptography’s Quantum-Safe Future Will Work

Lattice cryptography promises to protect secrets from the attacks of far-future quantum computers.

### Mathematicians Discover the Fibonacci Numbers Hiding in Strange Spaces

Recent explorations of unique geometric worlds reveal perplexing patterns, including the Fibonacci sequence and the golden ratio.

### 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.

### 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.

### Mathematical Connect-the-Dots Reveals How Structure Emerges

A new proof identifies precisely how large a mathematical graph must be before it contains a regular substructure.

### Unimaginable Surfaces Discovered After Decades-Long Search

Using ideas borrowed from graph theory, two mathematicians have shown that extremely complex surfaces are easy to traverse.

### How Complex Is a Knot? New Proof Reveals Ranking System That Works.

“Ribbon concordance” will let mathematicians compare knots by linking them across four-dimensional space.

### Mathematicians Coax Fluid Equations Into Nonphysical Solutions

The famed Navier-Stokes equations can lead to cases where more than one result is possible, but only in an extremely narrow set of situations.