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# Number theory

## Latest Articles

### Two Students Unravel a Widely Believed Math Conjecture

Mathematicians thought they were on the cusp of proving a conjecture about the ancient structures known as Apollonian circles. But a summer project would lead to its downfall.

### Elliptic Curves Yield Their Secrets in a New Number System

Ana Caraiani and James Newton have extended an important result in number theory to the imaginary realm.

### Why Mathematicians Re-Prove What They Already Know

It’s been known for thousands of years that the primes go on forever, but new proofs give fresh insights into how theorems depend on one another.

### New Proof Distinguishes Mysterious and Powerful ‘Modular Forms’

Using “refreshingly old” tools, mathematicians resolved a 50-year-old conjecture about how to categorize important functions called modular forms, with consequences for number theory and theoretical physics.

### Probability and Number Theory Collide — in a Moment

Mathematicians are taking ideas developed to study random numbers and applying them to a broad range of categories.

### The Year in Math

Four Fields Medals were awarded for major breakthroughs in geometry, combinatorics, statistical physics and number theory, even as mathematicians continued to wrestle with how computers are changing the discipline.

### Mathematical Trio Advances Centuries-Old Number Theory Problem

The work — the first-ever limit on how many whole numbers can be written as the sum of two cubed fractions — makes significant headway on “a recurring embarrassment for number theorists.”

### Teenager Solves Stubborn Riddle About Prime Number Look-Alikes

In his senior year of high school, Daniel Larsen proved a key theorem about Carmichael numbers — strange entities that mimic the primes.

### 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.