elliptic curves – Quanta Magazine https://www.quantamagazine.org Illuminating science Mon, 16 Sep 2024 14:10:47 -0400 en-US hourly 1 https://wordpress.org/?v=6.2 Elliptic Curve ‘Murmurations’ Found With AI Take Flight https://www.quantamagazine.org/elliptic-curve-murmurations-found-with-ai-take-flight-20240305/ https://www.quantamagazine.org/elliptic-curve-murmurations-found-with-ai-take-flight-20240305/#respond Tue, 05 Mar 2024 15:12:30 +0000 https://www.quantamagazine.org/?p=135959 The post Elliptic Curve ‘Murmurations’ Found With AI Take Flight first appeared on Quanta Magazine

]]> Elliptic curves are among the more beguiling objects in modern mathematics. They don’t seem complicated, but they form an expressway between the math that many people learn in high school and research mathematics at its most abstruse. They were central to Andrew Wiles’ celebrated proof of Fermat’s Last Theorem in the 1990s. They are key tools in modern cryptography. And in 2000...

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]]> https://www.quantamagazine.org/elliptic-curve-murmurations-found-with-ai-take-flight-20240305/feed/ 0 The Year in Math https://www.quantamagazine.org/the-biggest-discoveries-in-math-in-2023-20231222/ https://www.quantamagazine.org/the-biggest-discoveries-in-math-in-2023-20231222/#respond Sat, 23 Dec 2023 01:33:58 +0000 https://www.quantamagazine.org/?p=132809 The post The Year in Math first appeared on Quanta Magazine

]]> Mathematical truths are often born of the conflict between order and disorder. Mathematicians discover patterns, and, to better understand the mysterious forces at play, they look for countervailing impulses that disrupt those patterns. That tension came up repeatedly in our coverage this past year. We covered breakthroughs in graph theory, combinatorics, number theory and geometry — areas where...

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]]> https://www.quantamagazine.org/the-biggest-discoveries-in-math-in-2023-20231222/feed/ 0 The Astonishing Behavior of Recursive Sequences https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/ https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/#respond Thu, 16 Nov 2023 15:20:25 +0000 https://www.quantamagazine.org/?p=132363 The post The Astonishing Behavior of Recursive Sequences first appeared on Quanta Magazine

]]> In mathematics, simple rules can unlock universes of complexity and beauty. Take the famous Fibonacci sequence, which is defined as follows: It begins with 1 and 1, and each subsequent number is the sum of the previous two. The first few numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34 … Simple, yes, but this unassuming recipe gives rise to a pattern of far-reaching significance, one that appears to be...

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]]> https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/feed/ 0 Behold Modular Forms, the ‘Fifth Fundamental Operation’ of Math https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/ https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/#respond Thu, 21 Sep 2023 06:44:56 +0000 https://www.quantamagazine.org/?p=130315 The post Behold Modular Forms, the ‘Fifth Fundamental Operation’ of Math first appeared on Quanta Magazine

]]> “There are five fundamental operations in mathematics,” the German mathematician Martin Eichler supposedly said. “Addition, subtraction, multiplication, division and modular forms.” Part of the joke, of course, is that one of those is not like the others. Modular forms are much more complicated and enigmatic functions, and students don’t typically encounter them until graduate school. But “there...

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]]> https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/feed/ 0 Elliptic Curves Yield Their Secrets in a New Number System https://www.quantamagazine.org/elliptic-curves-yield-their-secrets-in-a-new-number-system-20230706/ https://www.quantamagazine.org/elliptic-curves-yield-their-secrets-in-a-new-number-system-20230706/#respond Thu, 06 Jul 2023 14:24:50 +0000 https://www.quantamagazine.org/?p=127581 The post Elliptic Curves Yield Their Secrets in a New Number System first appeared on Quanta Magazine

]]> Many complicated advances in research mathematics are spurred by a desire to understand some of the simplest questions about numbers. How are prime numbers distributed in the integers? Are there perfect cubes (like 8 = 23 or 27 = 33) that can be written as the sum of two other cubes? More generally, mathematicians might want to solve an equation. But it is often impossible to do so by tinkering...

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