Stories with Tag divisibility

• Decomposing a Factorial into Large Factors

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  Posted: 2025-03-28 14:55:54

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  Terry Tao explores the problem of efficiently decomposing a large factorial n! into a product of factors of roughly equal size √n. He outlines several approaches, including a naive iterative method that repeatedly divides n! by the largest integer below √n, and a more sophisticated approach leveraging prime factorization. The prime factorization method cleverly groups primes into products close to the target size, offering significant computational advantages. While both methods achieve the desired decomposition, the prime factorization technique highlights the interplay between the smooth structure of factorials (captured by their prime decomposition) and the goal of obtaining uniformly large factors. Tao emphasizes the efficiency gains from working with the prime factorization, and suggests potential generalizations and connections to other mathematical concepts like smooth numbers and the Dickman function.

  Terence Tao's blog post, "Decomposing a Factorial into Large Factors," explores the fascinating intersection of combinatorics and number theory by examining how to efficiently decompose a large factorial into a product of factors, each of which exceeds a given threshold. He begins by establishing the context of the problem, highlighting its relevance to scenarios such as distributing a large number of items (represented by the factorial) into containers subject to a minimum capacity constraint. This framework sets the stage for a deeper dive into the mathematical intricacies involved.

  Professor Tao then meticulously details an algorithmic approach to achieving this decomposition. The core of the algorithm revolves around a greedy strategy: iteratively extract the largest possible factor exceeding the prescribed threshold from the remaining factorial. This factor is determined by identifying the largest prime number less than or equal to the remaining factorial's square root and then computing the highest power of this prime that divides the remaining factorial. This power of the prime becomes a factor in the decomposition. This process is repeated, successively chipping away at the original factorial until the remaining portion falls below the specified threshold. The remaining portion, too small to be considered a "large factor," is then added as a final, possibly smaller, factor to complete the decomposition.

  The post proceeds to demonstrate the algorithm's efficacy through a concrete example, dissecting the decomposition of 100! with a minimum factor size of 10. This illustrative example provides a tangible understanding of the algorithm's mechanics. Furthermore, Tao discusses the efficiency of this greedy approach. While he stops short of rigorously proving optimality, he provides compelling intuitive arguments suggesting that this method yields a decomposition with a near-minimal number of factors. This resonates with the principle of greedily maximizing the size of each extracted factor, thereby minimizing the number of factors required to exhaust the original factorial.

  Finally, the post subtly touches upon the underlying number theory principles that underpin the algorithm's effectiveness. The process of identifying the largest prime power divisor leverages well-established number-theoretic concepts related to prime factorization and divisibility. While not explicitly delving into these theoretical underpinnings, the post subtly alludes to their importance in guaranteeing the algorithm's correctness and efficiency. In conclusion, Tao’s post provides a lucid and insightful exploration of an intriguing problem, offering a practical algorithm backed by intuitive explanations and connecting it to broader themes within number theory.

  • number theory
  • factorials
  • prime factorization
  • large factors
  • Decomposition
  • mathematics
  • Combinatorics
  • integer factorization
  • divisibility
  • primes

  Summary of Comments ( 4 )
  https://news.ycombinator.com/item?id=43506238

  Verbose tl;dr

  Hacker News users discussed the surprising difficulty of factoring large factorials, even when not seeking prime factorization. One commenter highlighted the connection to cryptography, pointing out that if factoring factorials were easy, breaking RSA would be as well. Another questioned the practical applications of this type of factorization, while others appreciated the mathematical puzzle aspect. The discussion also touched upon the computational complexity of factoring and the effectiveness of different factoring algorithms in this specific context. Some commenters shared resources and further reading on related topics in number theory. The general sentiment was one of appreciation for the mathematical curiosity presented by Terry Tao's blog post.

  The Hacker News post "Decomposing a Factorial into Large Factors," linking to a blog post by Terence Tao, has generated several comments discussing the mathematics involved and its potential applications.

  One commenter highlights the surprising nature of the result, expressing amazement that such a seemingly simple question about factorials could lead to a non-trivial mathematical problem. They appreciate the elegance of the proof and the unexpected connection to prime number theory.

  Another commenter delves into the computational aspects, pondering the efficiency of finding such decompositions. They discuss the possibility of using known factoring algorithms or if the specific structure of the problem allows for more specialized, faster techniques. The commenter also raises the question of whether there's a practical application for this decomposition, speculating on potential uses in cryptography or coding theory.

  A further comment focuses on the "smoothness" of the factors, referring to the property of having only small prime factors. They connect this to the concept of smooth numbers, which are frequently used in cryptographic algorithms, and suggest exploring the relationship further. This commenter also proposes investigating the distribution of the largest prime factor in the decomposition.

  One commenter with a username related to elliptic curves briefly mentions a potential link to elliptic curve cryptography, although they don't elaborate on the specific connection. This suggests a possible avenue for further exploration, hinting at a deeper relationship between the factorial decomposition and advanced cryptographic techniques.

  Another commenter points out the accessibility of Tao's explanation, praising his ability to break down complex mathematical concepts into understandable terms. They appreciate how the blog post makes the problem and its solution accessible to a wider audience.

  Finally, a commenter raises a more philosophical point about the nature of mathematical discovery. They reflect on how seemingly simple questions can lead to unexpected mathematical journeys and highlight the beauty of uncovering hidden connections between different areas of mathematics. They express admiration for Tao's ability to find and illuminate these connections.

  Overall, the comments on the Hacker News post demonstrate a mix of appreciation for the mathematical elegance of the result, curiosity about its computational implications, and speculation about potential applications. They also showcase the collaborative nature of the platform, where commenters build on each other's ideas and explore different facets of the topic.

• Where Do Those Undergraduate Divisibility Problems Come From?

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  Posted: 2025-01-20 09:41:39

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  The blog post explores the origin of seemingly arbitrary divisibility problems often encountered in undergraduate mathematics courses. It argues that these problems aren't typically plucked from thin air, but rather stem from broader mathematical concepts, particularly abstract algebra. The post uses the example of proving divisibility by 7 using a specific algorithm to illustrate how such problems can be derived from exploring properties of polynomial rings and quotient rings. Essentially, the apparently random divisibility rule is a consequence of working within a modular arithmetic system, which connects to deeper algebraic structures. The post aims to demystify these types of problems and show how they offer a glimpse into richer mathematical ideas.

  The blog post "Where Do Those Undergraduate Divisibility Problems Come From?" by Gil Grossack delves into the seemingly arbitrary nature of divisibility problems frequently encountered in undergraduate mathematics courses, particularly those involving prime factorization and modular arithmetic. Grossack argues that these problems often appear disconnected from any real-world application or deeper mathematical theory, leaving students to wonder about their origins and purpose. He proposes that many of these problems stem from the rich field of algebraic number theory, specifically the study of cyclotomic integers.

  The author begins by illustrating a typical undergraduate divisibility problem, showcasing the kind of question that might appear on a homework assignment or exam. He then meticulously deconstructs this example, revealing the underlying connection to cyclotomic integers and the properties of prime ideals in cyclotomic fields. Cyclotomic integers, defined as complex numbers that are roots of polynomials of the form xⁿ - 1, form a ring with intriguing divisibility properties. These properties, when examined through the lens of prime ideal factorization, provide the foundation for generating the seemingly ad-hoc divisibility problems often presented to undergraduate students.

  Grossack explains how the unique factorization of prime numbers in the ring of integers of a cyclotomic field translates into specific divisibility relationships involving ordinary integers. He meticulously traces the path from the abstract realm of cyclotomic fields to the concrete integer divisibility problems found in undergraduate textbooks. This connection, often obscured in introductory courses, illuminates the deeper mathematical structures underlying these seemingly simple problems.

  The post further elaborates on the concept of prime ideals and their role in understanding divisibility within cyclotomic fields. It explains how the factorization of prime ideals in these fields gives rise to congruences and divisibility relationships among ordinary integers. The author uses concrete examples and detailed explanations to demonstrate how specific divisibility problems can be derived from the properties of cyclotomic integers and prime ideals.

  In essence, the post argues that the seemingly arbitrary divisibility problems encountered by undergraduates are not merely contrived exercises but rather specific instances of more general and profound mathematical principles rooted in algebraic number theory. By exposing this underlying connection, Grossack aims to provide context and meaning to these problems, enriching the learning experience for students and demonstrating the interconnectedness of mathematical concepts. He concludes by suggesting that understanding the origins of these problems can transform them from rote exercises into windows into a richer mathematical landscape.

  • divisibility
  • mathematics
  • number theory
  • undergraduate education
  • math problems
  • Problem Solving
  • integer arithmetic
  • prime numbers
  • composite numbers
  • greatest common divisor
  • least common multiple
  • modular arithmetic
  • congruences
  • diophantine equations

  Summary of Comments ( 4 )
  https://news.ycombinator.com/item?id=42766825

  Verbose tl;dr

  The Hacker News comments discuss the origin and nature of "divisibility trick" problems often encountered in introductory number theory or math competitions. Several commenters point out that these problems often stem from exploring properties within modular arithmetic, even if not explicitly framed that way. Some suggest the problems are valuable for developing intuition about number systems and problem-solving skills. However, others argue that they can feel contrived or "magical," lacking connection to broader mathematical concepts. The idea of "casting out nines" is mentioned as a specific example, with some commenters highlighting its historical significance for checking calculations, while others dismiss it as a niche trick. A few commenters express a general appreciation for the linked blog post, praising its clarity and exploration of the topic.

  The Hacker News post "Where Do Those Undergraduate Divisibility Problems Come From?" with the ID 42766825 has several comments discussing the linked article about the origin of divisibility problems in undergraduate math courses.

  Several commenters appreciate the historical context provided by the article. One commenter notes the surprising connection between seemingly simple divisibility problems and deeper mathematical concepts like cyclotomic polynomials, remarking on how the article illuminates this link. They express enjoyment at learning the historical background of these problems.

  Another commenter focuses on the pedagogical implications of the article. They argue that presenting these problems with their historical context and motivation could make them more engaging for students. Instead of appearing as arbitrary exercises, understanding the problems' origins in Gaussian periods and cyclotomy could provide a deeper appreciation for their significance. This commenter believes that such an approach could enhance students' understanding and enjoyment of mathematics.

  A further commenter builds on this by suggesting that the article highlights a missed opportunity in math education. They posit that framing these problems within their historical context could transform them from rote exercises into explorations of deeper mathematical ideas. This, they argue, could foster a greater sense of curiosity and engagement in students.

  One commenter questions the article's assertion that these problems originate from Gauss's work on cyclotomy. They propose that the concepts of divisibility and modular arithmetic predate Gauss and likely arose independently in different cultures throughout history. They suggest that the article might overstate Gauss's contribution to the field while acknowledging that Gauss undoubtedly systematized and advanced these concepts. They also inquire about the historical usage of these problems in mathematical competitions.

  Another commenter offers a practical perspective, suggesting that many of these divisibility problems are chosen because they lend themselves well to being solved using modular arithmetic. They imply that this practicality, along with the ability to create problems with varying levels of difficulty, contributes to their prevalence in undergraduate courses.

  Finally, one commenter shares a personal anecdote about encountering similar problems during their own mathematical education. They mention being particularly fascinated by the problem of proving the divisibility rule for 7, showcasing the potential of these problems to spark curiosity and deeper exploration. They highlight that their interest stemmed not just from solving the problem but from understanding the underlying principles.