Stories with Tag number theory

• The three-dimensional Kakeya conjecture, after Wang and Zahl

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  Posted: 2025-02-27 16:53:56

  Verbose tl;dr

  Terry Tao's blog post discusses the recent proof of the three-dimensional Kakeya conjecture by Hong Wang and Joshua Zahl. The conjecture states that any subset of three-dimensional space containing a unit line segment in every direction must have Hausdorff dimension three. While previous work, including Tao's own, established lower bounds approaching three, Wang and Zahl definitively settled the conjecture. Their proof utilizes a refined multiscale analysis of the Kakeya set and leverages polynomial partitioning techniques, building upon earlier advances in incidence geometry. The post highlights the key ideas of the proof, emphasizing the clever combination of existing tools and innovative new arguments, while also acknowledging the remaining open questions in higher dimensions.

  Terence Tao's blog post, "The three-dimensional Kakeya conjecture, after Wang and Zahl," meticulously dissects the recent groundbreaking work of Hong Wang and Joshua Zahl, who have established a near-optimal bound for the dimension of Kakeya sets in three dimensions. The Kakeya conjecture, a long-standing problem in harmonic analysis with connections to numerous other mathematical fields, asserts that Kakeya sets, subsets of Euclidean space containing a unit line segment in every direction, must have full Hausdorff dimension. While the conjecture remains open in higher dimensions, Wang and Zahl's work represents a monumental leap forward in three dimensions, significantly improving upon previous estimates.

  Tao begins by meticulously defining the Kakeya conjecture and elucidating its significance, emphasizing its intricate relationship with problems in areas such as number theory, partial differential equations, and additive combinatorics. He highlights the historical context, outlining the incremental progress made over the years in attacking this challenging problem, and situates Wang and Zahl's achievement within this historical narrative. He explains how their work builds upon and refines earlier techniques, especially those related to the multilinear Kakeya inequality.

  The core of the post then delves into the technical intricacies of Wang and Zahl's proof. Tao painstakingly unpacks their innovative approach, which leverages a sophisticated interplay between geometric and combinatorial arguments. He elucidates their crucial use of the polynomial method, a powerful tool in incidence geometry, to control the interactions between tubes, the fundamental building blocks of Kakeya sets. He carefully explains how Wang and Zahl cleverly adapt and refine this method to the specific challenges posed by the three-dimensional Kakeya problem. Tao further details how they address the crucial issue of "stickiness," a phenomenon related to the potential overlap of tubes, by introducing a novel "graininess" parameter. This parameter allows them to more precisely quantify the interactions between tubes and thereby obtain sharper estimates.

  Furthermore, Tao elaborates on the concept of "bush" arguments, a key ingredient in previous approaches to the Kakeya problem. He explains how Wang and Zahl utilize a refined version of these arguments, incorporating their insights on stickiness and graininess to achieve a significant improvement in bounding the dimension of Kakeya sets. He emphasizes the delicate balance and intricate interplay between the various components of their proof.

  Finally, Tao concludes with a discussion of the remaining challenges in fully resolving the Kakeya conjecture, particularly in higher dimensions. He speculates on potential avenues for future research, suggesting that the techniques developed by Wang and Zahl could potentially be adapted and extended to higher-dimensional settings, offering a glimmer of hope for further progress on this long-standing open problem. He also highlights the broader implications of their work, underscoring its potential impact on other related problems in harmonic analysis and beyond.

  • mathematics
  • Kakeya Conjecture
  • Three-Dimensional Kakeya Conjecture
  • Harmonic Analysis
  • Geometric Measure Theory
  • number theory
  • Partial Differential Equations
  • Wang
  • Zahl
  • Terence Tao
  • Blog Post

  Summary of Comments ( 1 )
  https://news.ycombinator.com/item?id=43196110

  Verbose tl;dr

  HN commenters discuss the implications of the recent proof of the three-dimensional Kakeya conjecture, praising its elegance and accessibility even to non-experts. Several highlight the significance of "polynomial partitioning," the technique central to the proof, and its potential applications in other areas of mathematics. Some express excitement about the possibility of tackling higher dimensions, while others acknowledge the significant jump in complexity this would entail. The clear exposition of the proof by Tao is also commended, making the complex subject matter understandable to a broader audience. The connection to the original Kakeya needle problem and its surprising implications for analysis are also noted.

  The Hacker News post discussing Terry Tao's blog entry on the three-dimensional Kakeya conjecture has a modest number of comments, mostly focusing on the difficulty of the problem and the implications of the recent progress.

  One commenter highlights the significant challenge posed by the Kakeya conjecture, even in three dimensions, pointing out that while the problem might sound simple to a layperson, it has stumped mathematicians for decades. They express excitement at the new developments and the potential for further breakthroughs.

  Another comment emphasizes the intricate nature of the proof by Wang and Zahl, mentioning its length and complexity. They link this to the broader trend of increasingly complex proofs in advanced mathematics and the challenges this presents for verifying and understanding them. This comment also touches upon the use of computers in checking mathematical proofs, raising questions about the future role of computational tools in mathematical research.

  A further comment delves into the specifics of the Kakeya conjecture, explaining the concept of a "Besicovitch set" – a set containing a unit line segment in every direction but having arbitrarily small area. This comment helps to illustrate the counterintuitive nature of the problem and the difficulty in visualizing these sets.

  Another commenter draws a connection between the Kakeya conjecture and other open problems in mathematics, such as the Erdős distinct distances problem. They suggest that progress in one area can often lead to insights in seemingly unrelated fields, highlighting the interconnectedness of mathematical concepts.

  Finally, one comment focuses on Terry Tao's blog itself, praising its accessibility and ability to explain complex mathematical ideas to a broader audience. They appreciate Tao's efforts to break down difficult concepts into more digestible pieces, making the topic more approachable for non-experts.

  In summary, the comments on the Hacker News post reflect a general appreciation for the difficulty of the Kakeya conjecture, an excitement about the recent progress made by Wang and Zahl, and an interest in the broader implications for mathematics. They also highlight the value of clear explanations and the role of online platforms like Terry Tao's blog in disseminating complex mathematical ideas.

• Behold Modular Forms, the 'Fifth Fundamental Operation' of Math (2023)

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  Posted: 2025-02-24 18:04:50

  Verbose tl;dr

  Modular forms, complex functions with extraordinary symmetry, are revolutionizing how mathematicians approach fundamental problems. These functions, living in the complex plane's upper half, remain essentially unchanged even after being twisted and stretched in specific ways. This unusual resilience makes them powerful tools, weaving connections between seemingly disparate areas of math like number theory, analysis, and geometry. The article highlights their surprising utility, suggesting they act as a "fifth fundamental operation" akin to addition, subtraction, multiplication, and division, enabling mathematicians to perform calculations and uncover relationships previously inaccessible. Their influence extends to physics, notably string theory, and continues to expand mathematical horizons.

  The Quanta Magazine article, "Behold Modular Forms, the 'Fifth Fundamental Operation' of Math," explores the fascinating world of modular forms and their surprising connections across various mathematical domains. These complex mathematical objects, often described as possessing an almost supernatural level of symmetry, are functions defined on the complex upper half-plane. This means they operate on numbers that involve the imaginary unit i, the square root of -1. Their key characteristic is their invariance under a specific group of transformations known as modular transformations. These transformations, which involve fractional linear transformations, act like intricate rotations and reflections within this complex plane. The fact that modular forms remain essentially unchanged after these transformations is a testament to their deep-seated symmetry.

  This inherent symmetry isn't merely an aesthetic curiosity. It allows mathematicians to extract an astounding amount of information from these forms. The article likens this to squeezing an orange and obtaining far more juice than its size would suggest. This "extra" information manifests in various ways, including generating unexpected connections between seemingly disparate mathematical fields. The article highlights, for instance, the crucial role modular forms played in proving Fermat's Last Theorem, a centuries-old problem finally solved by Andrew Wiles in 1994. Wiles's proof hinged on demonstrating a link between elliptic curves, a type of algebraic equation, and modular forms, a connection previously conjectured but unproven.

  The article further emphasizes the idea of modular forms as representing a "fifth fundamental operation" in mathematics. Traditional arithmetic focuses on the four basic operations: addition, subtraction, multiplication, and division. Modular forms, however, introduce a new kind of operation, one that goes beyond simple manipulation of numbers. This "fifth operation" involves exploring and exploiting the intricate symmetries embedded within these forms. By examining how they transform and interact, mathematicians can unveil hidden relationships and structures within the mathematical landscape.

  The article delves into specific examples, including the j-invariant, a modular function (a slightly less constrained cousin of modular forms) that acts as a sort of "fingerprint" for elliptic curves. It maps every elliptic curve to a unique complex number, effectively classifying them. This ability to classify and organize complex mathematical objects is another hallmark of modular forms and highlights their organizational power.

  Furthermore, the article touches upon the burgeoning field of monstrous moonshine, which connects modular forms to the largest sporadic finite simple group, known as the Monster group. This unexpected link between seemingly abstract mathematical concepts hints at a deeper underlying structure within mathematics, a structure that modular forms are instrumental in revealing.

  In essence, the article presents modular forms not as isolated mathematical curiosities but as powerful tools that illuminate hidden connections and symmetries within the mathematical universe. They represent a shift in perspective, a move beyond traditional arithmetic to a more nuanced understanding of mathematical structure and relationships, effectively acting as a powerful lens through which to view the intricate tapestry of mathematics.

  • mathematics
  • Modular Forms
  • number theory
  • Abstract Algebra
  • Complex Analysis
  • Functions
  • Symmetry
  • Group Theory
  • Quanta Magazine
  • Math Research
  • Higher Mathematics
  • Elliptic Curves
  • Ramanujan
  • Analytic Number Theory

  Summary of Comments ( 2 )
  https://news.ycombinator.com/item?id=43162737

  Verbose tl;dr

  HN commenters generally expressed appreciation for the Quanta article's accessibility in explaining a complex mathematical concept. Several highlighted the connection between modular forms and both string theory and the monster group, emphasizing the unexpected bridges between seemingly disparate areas of math and physics. Some discussed the historical context of modular forms, including Ramanujan's contributions. A few more technically inclined commenters debated the appropriateness of the "fifth fundamental operation" phrasing, arguing that modular forms are more akin to functions or tools built upon existing operations rather than a fundamental operation themselves. The intuitive descriptions provided in the article were praised for helping readers grasp the core ideas without requiring deep mathematical background.

  The Hacker News post titled "Behold Modular Forms, the 'Fifth Fundamental Operation' of Math (2023)" has generated a moderate number of comments, with a significant portion focusing on clarifying the meaning of "fifth fundamental operation" and discussing the pedagogical implications of such a label.

  Several commenters express skepticism or disagreement with the characterization of modular forms as a fundamental operation. They point out that "operation" typically implies a procedure or action taken on mathematical objects, whereas modular forms are themselves mathematical objects (functions). One commenter suggests a more appropriate description would be "fundamental object of study" rather than "fundamental operation." Others humorously suggest alternative "fifth operations," such as exponentiation or tetration, highlighting the somewhat arbitrary nature of the claim.

  A recurring theme is the importance of context and target audience when using such terminology. Some commenters argue that the "fifth operation" label might be useful as a catchy hook for a popular science article, aimed at piquing the interest of a wider audience. However, they also acknowledge its potential to mislead or confuse readers, particularly those with a more formal mathematical background. One commenter specifically notes the challenge of teaching modular forms effectively to undergraduates, given their complexity.

  The discussion also touches upon the beauty and significance of modular forms in mathematics. Some commenters express appreciation for the article's attempt to explain a complex topic in an accessible way, while others delve into more technical aspects, mentioning connections to other areas of mathematics like number theory and cryptography. A few comments offer additional resources for those interested in learning more about modular forms, such as specific books and online courses.

  One commenter provides a nuanced perspective, suggesting that the "fifth operation" framing might refer to the modularity theorem, which demonstrates a profound connection between modular forms and elliptic curves. This connection, the commenter argues, could be considered a powerful "operation" that bridges different branches of mathematics.

  In summary, the comments largely grapple with the idea of modular forms as a "fifth fundamental operation," expressing varying degrees of skepticism, offering alternative interpretations, and highlighting the importance of context and audience. While acknowledging the article's attempt to make a complex topic more accessible, the comments also underscore the potential for such simplified terminology to be misleading. The discussion ultimately reflects a broader conversation about the nature of mathematical operations and the challenges of communicating sophisticated mathematical concepts to a wider audience.

• Making any integer with four 2s

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  Posted: 2025-02-23 15:24:02

  Verbose tl;dr

  The post explores the mathematical puzzle of representing any integer using four twos and a limited set of operations. It demonstrates how combining operations like addition, subtraction, multiplication, division, square roots, factorials, decimals, and concatenation, alongside techniques like logarithms and the gamma function (a generalization of the factorial), allows for expressing a wide range of integers. The author showcases examples and discusses the challenges of representing larger numbers, particularly prime numbers, due to the increasing complexity of the required expressions. The ultimate goal isn't a formal proof, but rather a practical exploration of the expressive power of combining these mathematical tools with a limited set of starting digits.

  This blog post, titled "Making any integer with four 2s," delves into the mathematical puzzle of representing any integer using exactly four instances of the digit 2 and a limited set of operations. The author meticulously explores various mathematical tools and techniques to achieve this goal. Starting with the basic arithmetic operations of addition, subtraction, multiplication, and division, the author demonstrates how combinations of these operations, along with the use of parentheses for enforcing order of operations, can generate a surprising number of integers.

  The author then introduces the concept of concatenation, which allows combining two twos to form 22, significantly expanding the range of achievable numbers. Further expanding the toolkit, the author incorporates the use of the square root, factorial, and decimal point (to create 2.2 or .2), illustrating how these additions further broaden the scope of constructible integers. For instance, the factorial operation allows for the creation of larger numbers like 22! or combinations like (2+2)!.

  The author acknowledges the potential ambiguity in interpreting concatenation and the decimal point and clarifies their intended usage within this puzzle's framework. To represent powers, the author employs the convention of repeated square roots, recognizing this as a slightly unconventional but effective representation within the constraints of the four-twos rule.

  The exploration goes beyond basic arithmetic operations, introducing the concept of logarithms, specifically base 10 and base 2 logarithms, represented as lg and lb respectively. The author carefully explains how logarithms can be used strategically in conjunction with the other allowed operations to generate specific integers, showcasing the power and versatility this mathematical function brings to the problem.

  The post demonstrates the process of constructing several specific integers as examples, highlighting the creativity and logical thinking required to manipulate the four twos into the desired result. It emphasizes the intricate interplay of different mathematical operations and how their clever combination can yield a wide range of integer values. The author concludes by inviting readers to explore and discover further possibilities within the established rules, encouraging continued engagement with this mathematical challenge.

  • mathematics
  • puzzles
  • number theory
  • Arithmetic
  • integers
  • Problem Solving
  • recreational mathematics
  • four twos
  • digital manipulation

  Summary of Comments ( 71 )
  https://news.ycombinator.com/item?id=43149883

  Verbose tl;dr

  HN commenters largely focused on the limitations and expansions of the puzzle. Some pointed out that the allowed operations weren't explicitly defined, leading to debates about the validity of certain solutions, particularly the use of the square root and floor/ceiling functions. Others discussed alternative approaches, such as using logarithms or the successor function. A few commenters explored variations of the puzzle, including using different numbers or a different quantity of the given number. The overall sentiment was one of intrigue, with many appreciating the puzzle's challenge and the creativity it sparked.

  The Hacker News post titled "Making any integer with four 2s" has a modest number of comments, mostly focusing on variations of the puzzle and different mathematical approaches.

  One commenter points out the ambiguity of the original problem statement regarding allowed operations. They clarify that the standard interpretation permits the use of basic arithmetic operations (addition, subtraction, multiplication, division), square roots, factorials, concatenation, decimal points, and sometimes powers and logs. They also suggest the use of the "binary concatenation operator", using .2 to represent 0.2 and creating arbitrary binary numbers, although acknowledging this might be considered "cheating."

  Another commenter discusses the challenge of creating the number 7, mentioning strategies involving floor and ceiling functions applied to square roots and logs, ultimately leading to complex expressions. This commenter also highlights the importance of the specific allowed operations in determining the solvability of the puzzle for certain numbers.

  The concept of the "four fours" puzzle is brought up, referencing a book ("Mathematical Recreations and Essays" by W. W. Rouse Ball) that discusses the puzzle and its variations. This commenter also suggests that the "four twos" version is likely more challenging.

  Another comment thread discusses the importance of clearly defining the set of allowed operations, with one commenter specifically showing how to generate the numbers 1 through 10 using a relatively standard set of operations (including square root, factorial, concatenation, the floor function, and exponentiation).

  Finally, one commenter introduces the concept of allowing the use of an infinite number of square roots, potentially simplifying the problem considerably, though deviating significantly from the typical rules.

• Making any integer with four 2s

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  Posted: 2025-02-23 02:25:51

  Verbose tl;dr

  The post explores the mathematical puzzle of representing any integer using four twos and a limited set of operations. It demonstrates how combining operations like addition, subtraction, multiplication, division, square roots, factorials, decimal points, and concatenation, along with concepts like double factorials and the gamma function (a generalization of the factorial), allows for creative expression of numerous integers. While acknowledging the potential for more complex representations using less common operations, the post focuses on showcasing the flexibility and surprising reach of this mathematical exercise using a relatively small toolkit of functions. It ultimately highlights the challenge and ingenuity involved in manipulating a limited set of numbers to achieve a wide range of results.

  This blog post delves into the fascinating mathematical puzzle of representing any given integer using exactly four instances of the digit '2', along with a variety of permitted mathematical operations. The author meticulously outlines the permissible operations, which include the fundamental arithmetic operations of addition, subtraction, multiplication, and division. Furthermore, the author allows for the use of exponentiation, square roots (represented by the conventional radical symbol), factorials, concatenation (joining the digits together, e.g., forming 22 from two 2s), and decimal points (to create numbers like 2.2). Parentheses are also permitted to explicitly control the order of operations and enforce desired groupings.

  The central challenge, as articulated by the author, is to devise expressions using precisely four 2s and the allowed operations to generate each integer sequentially. The post then proceeds to demonstrate solutions for a range of integers, starting from 0 and continuing upwards. These demonstrations showcase the ingenious application of various combinations of the allowed operations. For example, the integer 0 is represented as (2 + 2 - 2 - 2). The author highlights the escalating complexity involved in representing larger integers, illustrating how cleverly combining operations becomes crucial to achieve the desired results.

  The post emphasizes the explorative and often counterintuitive nature of the problem-solving process. It highlights the satisfaction derived from discovering elegant and sometimes unexpected solutions. The author does not claim to provide a generalized formula or algorithm for solving the puzzle for any given integer. Rather, the post focuses on showcasing specific examples and demonstrating the kind of creative mathematical thinking required to tackle such challenges. The ultimate objective isn't necessarily to find all possible representations for each integer, but rather to demonstrate the feasibility of representing a variety of integers within the imposed constraints of using only four 2s and the specified operations.

  • mathematics
  • puzzles
  • number theory
  • Arithmetic
  • integers
  • recreational mathematics
  • Problem Solving
  • four twos
  • 2s
  • challenge

  Summary of Comments ( 140 )
  https://news.ycombinator.com/item?id=43145753

  Verbose tl;dr

  Hacker News users generally enjoyed the puzzle presented in the linked article about constructing integers using four twos. Several commenters explored alternative solutions using different mathematical operations like bitwise XOR, square roots, and logarithms, showcasing a playful engagement with the challenge. Some discussed the arbitrary nature of the "four twos" constraint, suggesting that similar puzzles could be devised with other numbers or constraints. A few comments delved into the role of such puzzles in education, highlighting their value in encouraging creative problem-solving. One commenter pointed out the similarity to the "four fours" puzzle, referencing a website dedicated to exploring its variations.

  The Hacker News post "Making any integer with four 2s" has generated several comments discussing various aspects of the mathematical puzzle presented in the linked article. The comments largely revolve around alternative solutions, generalizations of the puzzle, and debates about the "legality" of certain operations.

  One commenter points out that the puzzle allows for a broader range of operations than typically intended, making it relatively easy to generate any integer. They specifically mention the use of the square root and floor function, which allows for the creation of 1 (sqrt(2)/sqrt(2) = 1) and 0 (floor(sqrt(2)/22) = 0), and from there any other integer. This commenter also notes the lack of strict rules surrounding concatenation, which can be exploited further.

  Another commenter focuses on the use of the logarithm function, suggesting a more generalized approach. They propose using log base 2 of the square root of 2 (log₂(√2)) to obtain 1/2, and subsequently generating other fractions and integers. This comment sparks a discussion about which operations are "acceptable" within the puzzle's framework, with some arguing that logarithms are too powerful and stretch the intended spirit of the challenge.

  The discussion also touches upon the concept of using the successor function, denoted as S(n) = n+1, though this is generally dismissed as being outside the usual scope of such puzzles. A similar sentiment is expressed regarding the use of infinite series, which are considered too powerful and make the puzzle trivial.

  Several commenters express their enjoyment of the puzzle and explore alternative solutions using more standard operations like addition, subtraction, multiplication, division, square roots, and factorials. Some commenters propose restricting the allowed operations to make the puzzle more challenging.

  A significant part of the discussion centers on the ambiguity of the rules, specifically regarding concatenation and the allowed set of operations. This ambiguity leads to creative solutions but also to debates on whether those solutions adhere to the (unstated) intended rules of the puzzle. Some commenters suggest that explicitly defining the allowed operations would lead to a more focused and interesting challenge.

• After 20 Years, Math Couple Solves Major Group Theory Problem

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  Posted: 2025-02-20 10:09:56

  Verbose tl;dr

  Mathematicians and married couple, George Willis and Monica Nevins, have solved a long-standing problem in group theory concerning just-infinite groups. After two decades of collaborative effort, they proved that such groups, which are infinite but become finite when any element is removed, always arise from a specific type of construction related to branch groups. This confirms a conjecture formulated in the 1990s and deepens our understanding of the structure of infinite groups. Their proof, praised for its elegance and clarity, relies on a clever simplification of the problem and represents a significant advancement in the field.

  In a remarkable testament to persistence and collaborative ingenuity, mathematicians Martin Liebeck and Jan Saxl, a married couple residing in London, have successfully resolved a long-standing conundrum within the intricate field of group theory, a branch of abstract algebra exploring the properties of symmetry. This significant achievement, culminating after two decades of dedicated research, centers on the classification of maximal subgroups of finite almost simple groups, a concept essential for comprehending the complex structures and interrelationships within these fundamental mathematical objects.

  Specifically, Liebeck and Saxl's work provides a comprehensive description of the maximal subgroups of exceptional groups of Lie type, intricate algebraic constructs named after the 19th-century Norwegian mathematician Sophus Lie. These exceptional groups, designated E6, E7, E8, F4, and G2, represent a fascinating class of structures that resist easy categorization and have proven particularly challenging to analyze. Prior to this breakthrough, a complete understanding of their maximal subgroups remained elusive, hindering a deeper understanding of the groups themselves and their connections to other areas of mathematics and physics.

  The breakthrough achieved by Liebeck and Saxl solidifies the monumental project of classifying all finite simple groups, often referred to as the "Enormous Theorem." This enormously complex undertaking, spanning decades and involving the collaborative efforts of numerous mathematicians worldwide, aimed to categorize the fundamental building blocks of symmetry. The classification of maximal subgroups is a crucial extension of this endeavor, as it elucidates how larger groups are constructed from smaller, simpler components.

  Their work, drawing on a wide array of advanced mathematical tools and techniques, involved meticulous analysis of the intricate web of relationships between subgroups within these exceptional groups. This involved carefully distinguishing between different types of subgroups, classifying them according to their properties, and establishing rigorous bounds on their size and complexity. The results offer a crucial advancement in understanding the internal architecture of exceptional Lie groups, laying the foundation for future research in group theory and its diverse applications.

  The collaborative nature of this accomplishment is also noteworthy, highlighting the power of shared intellectual pursuit. As a married couple, Liebeck and Saxl have maintained a long-standing professional partnership, combining their individual expertise and insights to tackle this intricate problem. Their combined efforts have not only resolved a significant mathematical question but also showcase the enriching potential of collaborative research. Their achievement stands as a testament to the dedication, creativity, and perseverance required to unravel the profound mysteries of abstract mathematics.

  • mathematics
  • Group Theory
  • Problem Solving
  • Research
  • collaboration
  • Mathematicians
  • Science
  • academia
  • Higher Education
  • number theory
  • Abstract Algebra

  Summary of Comments ( 50 )
  https://news.ycombinator.com/item?id=43113024

  Verbose tl;dr

  Hacker News commenters generally expressed awe and appreciation for the mathematicians' dedication and the elegance of the solution. Several highlighted the collaborative nature of the work and the importance of such partnerships in research. Some discussed the challenge of explaining complex mathematical concepts to a lay audience, while others pondered the practical applications of this seemingly abstract work. A few commenters with mathematical backgrounds offered deeper insights into the proof and its implications, pointing out the use of representation theory and the significance of classifying groups. One compelling comment mentioned the personal connection between Geoff Robinson and the commenter's advisor, offering a glimpse into the human side of the mathematical community. Another interesting comment thread explored the role of intuition and persistence in mathematical discovery, highlighting the "aha" moment described in the article.

  The Hacker News post titled "After 20 Years, Math Couple Solves Major Group Theory Problem" has generated a modest number of comments, focusing primarily on the human aspect of the achievement and touching upon the implications of the solved problem. No one delves deep into the mathematical specifics, likely due to the complexity of the topic.

  Several comments highlight the remarkable dedication and collaborative spirit of the mathematicians involved. One commenter expresses admiration for the couple's persistence over two decades, emphasizing the significance of their partnership in achieving this breakthrough. This sentiment is echoed by another comment which celebrates their joint effort as a heartwarming testament to collaborative problem-solving.

  Another thread discusses the nature of "finally solved" in the context of mathematics, acknowledging that the solution itself might open up new avenues of inquiry and further complexities. This comment suggests that the solution, while significant, may not represent a definitive end but rather a stepping stone to further mathematical exploration.

  A few comments mention the application of group theory in various fields like cryptography and particle physics, albeit without going into specific details. These comments serve to briefly contextualize the relevance of the achievement within a broader scientific landscape.

  One commenter expresses a desire for a more accessible explanation of the problem and its solution, highlighting the challenge of communicating complex mathematical concepts to a wider audience. This comment underscores the inherent difficulty in bridging the gap between specialized mathematical research and public understanding.

  In summary, the comments on the Hacker News post primarily focus on the dedication and collaborative spirit of the mathematicians, the implications of solving a long-standing problem, and the broader relevance of group theory. The technical details of the solution are not discussed, likely due to the complexity of the subject matter. The overall tone is one of appreciation for the achievement and curiosity about its implications.

• Where Do Those Undergraduate Divisibility Problems Come From?

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

  Verbose tl;dr

  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

  Verbose tl;dr

  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.