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number theory

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A complex machine produces prime numbers onto a conveyor belt that winds backwards into infinity.
number theory

Why Mathematicians Re-Prove What They Already Know

By Anna Kramer
April 26, 2023
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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.

A modular form is shown as a distorted, rainbow-hued dome atop concentric circles.
number theory

New Proof Distinguishes Mysterious and Powerful ‘Modular Forms’

By Jordana Cepelewicz
March 9, 2023
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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.

number theory

Probability and Number Theory Collide — in a Moment

By Leila Sloman
January 12, 2023
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Mathematicians are taking ideas developed to study random numbers and applying them to a broad range of categories.

2022 in Review

The Year in Math

By Konstantin Kakaes
December 22, 2022
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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.

number theory

Mathematical Trio Advances Centuries-Old Number Theory Problem

By Erica Klarreich
November 29, 2022
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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.”

A teenager wearing a blue polo shirt and glasses.
number theory

Teenager Solves Stubborn Riddle About Prime Number Look-Alikes

By Jordana Cepelewicz
October 13, 2022
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In his senior year of high school, Daniel Larsen proved a key theorem about Carmichael numbers — strange entities that mimic the primes.

number theory

A Numerical Mystery From the 19th Century Finally Gets Solved

By Leila Sloman
August 15, 2022
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Two mathematicians have proven Patterson’s conjecture, which was designed to explain a strange pattern in sums involving prime numbers.

An illustration of a large, 3D equation that almost looks like a sculpture (that is being inspected by different people, and is surrounded by different numbers and measuring equipment such as rulers and protractors. The equation is x2 − dy2 = –1).
number theory

Mathematicians Crack a Simple but Stubborn Class of Equations

By Jordana Cepelewicz
August 10, 2022
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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.

A deltoid and other mathematical shapes.
geometry

A Question About a Rotating Line Helps Reveal What Makes Real Numbers Special

By Kevin Hartnett
July 26, 2022
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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.


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