Stories with Tag prime numbers

• 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.

• Prime Numbers So Memorable That People Hunt for Them

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  Posted: 2025-01-18 14:41:53

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  Certain prime numbers possess aesthetically pleasing or curious properties that make them stand out and become targets for "prime hunters." These include palindromic primes (reading the same forwards and backwards), repunit primes (consisting only of the digit 1), and Mersenne primes (one less than a power of two). The rarity and mathematical beauty of these special primes drive both amateur and professional mathematicians to seek them out using sophisticated algorithms and distributed computing projects, pushing the boundaries of computational power and our understanding of prime number distribution.

  Within the fascinating realm of number theory, a peculiar subset of prime numbers, known as "memorable primes," has captured the attention and sparked the dedicated pursuit of both amateur mathematicians and seasoned professionals. These numerical gems, distinguished by their easily recallable and aesthetically pleasing digit sequences, possess a certain allure that transcends their purely mathematical significance. The article "Prime Numbers So Memorable That People Hunt for Them," published in Scientific American, delves into this intriguing phenomenon, exploring the motivations and methodologies employed in the search for these elusive numerical treasures.

  The article commences by elucidating the fundamental nature of prime numbers—those divisible only by one and themselves—and their crucial role in various mathematical disciplines, including cryptography. It then introduces the concept of memorable primes, categorizing them into several distinct families. These categories encompass repdigit primes, composed solely of a repeating digit; palindromic primes, which read the same forwards and backwards; and prime numbers exhibiting specific patterns, such as ascending or descending sequences of digits.

  The article subsequently elaborates on the various techniques utilized by "prime hunters" in their quest to discover new memorable primes. These methods range from employing sophisticated computer algorithms and distributed computing projects to leveraging theoretical insights from number theory. The article highlights the significant computational resources required to identify increasingly large prime numbers, particularly those exhibiting specific patterns. This computational challenge, in turn, contributes to the sense of accomplishment and recognition associated with discovering a new memorable prime.

  Furthermore, the article delves into the psychological aspects of this pursuit, suggesting that the inherent human tendency to seek patterns and recognize order within apparent chaos plays a significant role in the fascination with memorable primes. The ease with which these primes can be recalled and recognized contributes to their perceived aesthetic appeal. This intrinsic allure, combined with the intellectual challenge of their discovery, fosters a sense of satisfaction and intellectual stimulation among those engaged in the hunt.

  The article concludes by emphasizing the ongoing nature of this numerical exploration, highlighting the fact that countless memorable primes likely remain undiscovered, awaiting the diligent efforts of future prime hunters. It underscores the collaborative nature of this endeavor, with mathematicians and enthusiasts across the globe contributing to the ever-expanding catalog of these remarkable numerical entities. Ultimately, the article portrays the search for memorable primes not merely as a mathematical exercise, but as a testament to human curiosity, ingenuity, and the enduring fascination with the intricate beauty of the numerical world.

  • prime numbers
  • mathematics
  • number theory
  • memorable primes
  • prime hunting
  • recreational mathematics
  • pattern recognition
  • digit patterns
  • Scientific American

  Summary of Comments ( 47 )
  https://news.ycombinator.com/item?id=42748691

  Verbose tl;dr

  HN commenters largely discussed the memorability and aesthetics of the listed prime numbers, debating whether the criteria truly made them special or just reflected pattern-seeking tendencies. Some questioned the article's focus on base 10 representation, arguing that memorability is subjective and base-dependent. Others appreciated the exploration of mathematical beauty and shared their own favorite "interesting" numbers. Several commenters noted the connection to Smarandache sequences and other recreational math concepts, with links provided for further exploration. The practicality of searching for such primes was also questioned, with some suggesting it was merely a curiosity with no real-world application.

  The Hacker News post titled "Prime Numbers So Memorable That People Hunt for Them" (linking to a Scientific American article about memorable primes) has generated several comments, mostly engaging with the concept and offering further examples or related ideas.

  Several commenters discuss the memorability and aesthetics of different bases besides base-10. One commenter suggests that base-6 would be interesting, highlighting how palindromic primes might become more common. Another agrees, adding that divisibility rules would become more apparent in base-6. This thread extends with a playful thought experiment about how aliens with six fingers might find base-6 more intuitive.

  Another significant thread focuses on the practicality and application of such "memorable" primes. One commenter questions the actual usefulness of easily memorizable primes, expressing skepticism about any real-world application beyond recreational mathematics. Another commenter counters this by suggesting potential use in cryptography or as checksums, arguing that memorability could be an advantage in specific niche scenarios. This thread branches out into a discussion about the feasibility and security implications of using such primes for cryptographic purposes, with some skepticism expressed about their vulnerability due to their easily recognizable patterns.

  Several comments provide additional examples of interesting primes, such as those that are palindromic, or those that exhibit repeating digit patterns. One commenter points out the existence of programs and websites dedicated to finding and listing these kinds of primes, reflecting the ongoing interest in their discovery.

  One commenter mentions a tangential but related topic: the memorization of mathematical constants like Pi, pointing to online resources that aid in this endeavor. This leads to a brief discussion about mnemonics and memory techniques in general.

  A few comments simply express appreciation for the article and the concept of aesthetically pleasing primes, highlighting the intersection of mathematics and aesthetics.

  Finally, there's a short exchange discussing the prevalence of certain digits in different number bases and the potential reasons behind those distributions. This drifts slightly from the main topic but adds another layer of mathematical curiosity to the conversation.