integers – Quanta Magazine

How the Square Root of 2 Became a Number

Jordana Cepelewicz — Fri, 21 Jun 2024 13:50:15 +0000

The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction. The first proof of this...

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The Hidden Connection That Changed Number Theory

Max G. Levy — Wed, 01 Nov 2023 14:37:03 +0000

There are three kinds of prime numbers. The first is a solitary outlier: 2, the only even prime. After that, half the primes leave a remainder of 1 when divided by 4. The other half leave a remainder of 3. (5 and 13 fall in the first camp, 7 and 11 in the second.) There is no obvious reason that remainder-1 primes and remainder-3 primes should behave in fundamentally different ways. But they do.

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Two Students Unravel a Widely Believed Math Conjecture

Max G. Levy — Thu, 10 Aug 2023 14:49:03 +0000

Summer Haag and Clyde Kertzer had high hopes for their summer research project. Blindsiding an entire subfield of mathematics was not one of them. In May, Haag was finishing her first year of graduate school at the University of Colorado, Boulder, where Kertzer was an undergraduate. Both looked forward to a break from classes. Haag planned to explore new hikes and climbing routes. Kertzer...

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Mathematicians Crack a Simple but Stubborn Class of Equations

Jordana Cepelewicz — Wed, 10 Aug 2022 16:14:46 +0000

In the third century BCE, Archimedes posed a riddle about herding cattle that, he claimed, only a truly wise person could solve. His problem ultimately boiled down to an equation that involves the difference between two squared terms, which can be written as x2 – dy2 = 1. Here, d is an integer — a positive or negative counting number — and Archimedes was looking for solutions where both x and y...

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Math’s ‘Oldest Problem Ever’ Gets a New Answer

Jordana Cepelewicz — Wed, 09 Mar 2022 12:55:13 +0000

Number theorists are always looking for hidden structure. And when confronted by a numerical pattern that seems unavoidable, they test its mettle, trying hard — and often failing — to devise situations in which a given pattern cannot appear. One of the latest results to demonstrate the resilience of such patterns, by Thomas Bloom of the University of Oxford, answers a question with roots that...

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