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# number theory

## Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced

Decades after the landmark proof of Fermat’s Last Theorem, ideas abound for how to make it even more reliable. But such efforts reflect a deep misunderstanding of what makes the proof so important.

## The Subtle Art of the Mathematical Conjecture

It’s an educated guess, not a proof. But a good conjecture will guide math forward, pointing the way into the mathematical unknown.

## Mathematicians Discover the Perfect Way to Multiply

By chopping up large numbers into smaller ones, researchers have rewritten a fundamental mathematical speed limit.

## Sum-of-Three-Cubes Problem Solved for ‘Stubborn’ Number 33

A number theorist with programming prowess has found a solution to 33 = x³ + y³ + z³, a much-studied equation that went unsolved for 64 years.

## Smaller Is Better: Why Finite Number Systems Pack More Punch

Recent progress on the “sum product” problem recalls a celebrated mathematical result that revealed the power of miniature number systems.

## How a Strange Grid Reveals Hidden Connections Between Simple Numbers

A graduate student has helped illuminate a long-suspected connection between addition and multiplication.

## Mathematicians Seal Back Door to Breaking RSA Encryption

Digital security depends on the difficulty of factoring large numbers. A new proof shows why one method for breaking digital encryption won’t work.

## In the Universe of Equations, Virtually All Are Prime

Equations, like numbers, cannot always be split into simpler elements.

## New Proof Shows Infinite Curves Come in Two Types

Alexander Smith’s work on the Goldfeld conjecture reveals fundamental characteristics of elliptic curves.