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Category: Mathematics

Kissing Number Breakthrough: A New Approach to an Old Problem

2025-01-16
Kissing Number Breakthrough: A New Approach to an Old Problem

For over three centuries, mathematicians have grappled with the kissing number problem: how many identical spheres can touch a central sphere without overlapping? While the answer is 12 in three dimensions, higher dimensions remain a mystery. Recently, MIT undergraduate Anqi Li and Professor Henry Cohn devised a novel approach, abandoning traditional symmetry assumptions. Their unconventional, asymmetric strategy improved estimates for the kissing number in dimensions 17 through 21, marking the first progress in these dimensions since the 1960s. This breakthrough challenges established methods based on information theory and error-correcting codes, opening new avenues for solving this enduring mathematical puzzle.

(www.quantamagazine.org)
Mathematics Geometry Higher Dimensions

Can We Understand This Proof? A Glimpse into Formalized Mathematics

2025-01-10
Can We Understand This Proof? A Glimpse into Formalized Mathematics

Stephen Wolfram delves into a long-standing mathematical puzzle: the proof of a surprisingly simple axiom for Boolean algebra. Generated using automated theorem proving, the proof is incredibly complex and remains incomprehensible to humans. The article explores the proof's intricacies, dissecting its 'machine code' level operations, and poses a challenge: to humanize this proof. It discusses the potential of Large Language Models (LLMs) to understand and simplify the proof, and the implications for the future of mathematics. The conclusion suggests that some mathematical proofs may inherently be uninterpretable, hinting that mathematics will increasingly resemble an experimental science.

(writings.stephenwolfram.com)
Mathematics automated theorem proving formalized mathematics computational irreducibility

Multiplicative Infinitesimals: A New Calculus Approach

2025-01-08
Multiplicative Infinitesimals: A New Calculus Approach

This paper introduces a new concept called "multiplicative infinitesimals," analogous to traditional additive infinitesimals, to construct a new calculus system. Unlike traditional calculus based on differences, multiplicative calculus is based on quotients, using a Leibniz-like notation but with 'q' instead of 'd', representing a multiplicative perturbation of an expression. The author establishes the relationship between 'q' and 'd' through logarithmic and exponential operations and applies it to elasticity theory and multiplicative derivative calculations. This approach may offer new solutions to problems intractable with traditional methods.

(github.com)
Mathematics calculus infinitesimals

Mathematicians Discover New Way to Count Prime Numbers

2024-12-13
Mathematicians Discover New Way to Count Prime Numbers

Mathematicians Ben Green and Mehtaab Sawhney have proven there are infinitely many prime numbers of the form p² + 4q², where p and q are also primes. Their proof ingeniously utilizes Gowers norms, a tool from a different area of mathematics, demonstrating its surprising power in prime number counting. This breakthrough deepens our understanding of prime number distribution and opens new avenues for future research.

(www.quantamagazine.org)
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Mathematics prime numbers number theory Gowers norms