Stories with Tag prime numbers

• Show HN: New world record – verified Goldbach Conjecture up to 4*10^18+7*10^13

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  Posted: 2025-04-19 06:11:37

  Verbose tl;dr

  A distributed computing project leveraging idle CPU time from volunteers' computers has set a new verification record for the Goldbach Conjecture. The project, utilizing a novel grid computing approach, has confirmed the conjecture – which states that every even number greater than 2 can be expressed as the sum of two primes – up to 4 * 10^18 + 7 * 10^13. This surpasses previous verification efforts by a significant margin and demonstrates the potential of harnessing distributed computing power for tackling complex mathematical problems.

  A significant advancement in verifying the Goldbach Conjecture has been achieved, establishing a new world record. The Goldbach Conjecture, a prominent unsolved problem in number theory, posits that every even integer greater than 2 can be expressed as the sum of two prime numbers. While a complete proof remains elusive, computational verification has been pursued to test the conjecture's validity up to increasingly large numbers. This new record extends the verification to an impressive upper bound of 4 * 10^18 plus an additional 7 * 10^13.

  This achievement was accomplished through a distributed grid computing approach. The project, known as the "GridBach Project," harnessed the power of thousands of volunteers donating their unused computing resources. By dividing the immense computational task into smaller, manageable chunks distributed across this global network of computers, the project was able to achieve a scale of computation previously unattainable. This distributed approach allowed for parallel processing, dramatically accelerating the verification process.

  The prior record, while impressive in its own right, was significantly surpassed by this new milestone. The extension represents a substantial leap forward in the computational verification of the Goldbach Conjecture, providing further empirical support for the conjecture's truth while simultaneously highlighting the power of distributed computing in tackling complex mathematical problems. The project meticulously tracked the progress of each individual computation and employed robust verification mechanisms to ensure the accuracy and reliability of the results. This rigor underscores the validity of the new record and reinforces the significance of this contribution to the ongoing effort to understand the Goldbach Conjecture.

  • mathematics
  • number theory
  • goldbach conjecture
  • Distributed Computing
  • grid computing
  • world record
  • verification
  • Algorithm
  • prime numbers

  Summary of Comments ( 85 )
  https://news.ycombinator.com/item?id=43734583

  Verbose tl;dr

  Hacker News users discuss the computational resources used for the Goldbach conjecture verification, questioning the value and novelty of the achievement. Some commenters express skepticism about the significance of extending the verification limit, arguing that it doesn't contribute significantly to proving the conjecture itself. Others point out the inefficiency of the distributed grid computing approach compared to more optimized single-machine implementations. A few users discuss the specific hardware and software used in the project, including the use of BOINC and GPUs, while others debate the proper way to credit contributors in such distributed projects. Several commenters express concern about the lack of available source code and details on the verification methodology, hindering independent verification and analysis.

  The Hacker News post discussing the new world record for verifying Goldbach's Conjecture has a modest number of comments, mostly focusing on the technical aspects of the distributed computing approach used and the nature of the conjecture itself.

  Several commenters delve into the specifics of the grid computing system employed. One user questions the efficiency gains of this distributed approach compared to utilizing a single, powerful machine, highlighting potential overheads associated with network communication and data transfer. Another commenter speculates on the possibility of optimizing the verification process further by leveraging SIMD (Single Instruction, Multiple Data) instructions, potentially leading to even faster computation times. There's also a brief discussion regarding the memory requirements of such an endeavor, with one commenter suggesting that RAM limitations wouldn't be a major hurdle.

  Another thread of discussion revolves around the mathematical implications of the Goldbach Conjecture and the nature of "proof" versus "verification." One commenter points out that while the project provides further strong evidence supporting the conjecture, it doesn't constitute a mathematical proof. They elaborate on the difference between verifying the conjecture up to a certain limit and proving it for all even numbers greater than 2. Another user concurs, adding that despite the impressive scale of the verification, it remains "an interesting data point, not a mathematical breakthrough."

  A few comments address the practicalities of the project. One user asks about the availability of the source code, indicating an interest in examining the implementation details. Another commenter questions the overall value of the project, expressing skepticism about the scientific merit of merely pushing the verification limit higher.

  Finally, there are some brief exchanges regarding the history of the Goldbach Conjecture and previous attempts to verify it. One commenter mentions a prior effort using BOINC (Berkeley Open Infrastructure for Network Computing) and inquires about the differences between that project and the one discussed in the article.

  In summary, the comments section provides a mix of technical insights into the distributed computing aspect of the project, discussions about the mathematical nature of the Goldbach Conjecture, and some pragmatic questions regarding the project's implementation and significance. While there isn't a single overwhelmingly compelling comment, the collective discussion offers a nuanced perspective on the achievement and its limitations.

• Fibonacci Hashing: The Optimization That the World Forgot

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  Posted: 2025-04-14 01:02:41

  Verbose tl;dr

  Fibonacci hashing offers a faster alternative to the typical modulo operator (%) for distributing items into hash tables, especially when the table size is a power of two. It leverages the golden ratio's properties by multiplying the hash key by a large constant derived from the golden ratio and then bit-shifting the result, effectively achieving a modulo operation without the expensive division. This method produces a more even distribution compared to modulo with prime table sizes, particularly when dealing with keys exhibiting sequential patterns, thus reducing collisions and improving performance. While theoretically superior, its benefits may be negligible in modern systems due to compiler optimizations and branch prediction for modulo with powers of two.

  The blog post "Fibonacci Hashing: The Optimization That the World Forgot (or a Better Alternative to Integer Modulo)" by Christopher Wellons explores a highly efficient hashing technique based on the golden ratio, arguing that it's often superior to the commonly used modulo operator for distributing hash values across a hash table. Wellons begins by explaining the shortcomings of the modulo operator, particularly when the table size is not a prime number. If the table size has common factors with the hash values, the modulo operation can lead to clustering and reduced performance. This is because the modulo will effectively only distribute the keys among a subset of the available slots, proportional to the greatest common divisor of the table size and the hash.

  He then introduces the concept of Fibonacci hashing, which utilizes a specific multiplication and bitwise shift operation as a replacement for modulo. This technique relies on the properties of the golden ratio, an irrational number closely approximated by the ratio of consecutive Fibonacci numbers. The golden ratio's inherent connection to relatively prime numbers allows for more even distribution of hash values even when the table size is not prime, and especially when it’s a power of two. This is achieved by multiplying the hash value by a large integer representation of the golden ratio's fractional part (specifically 2⁶⁴ * φ_f where φ_f is the fractional part of the golden ratio) and then taking the high bits of the result, equivalent to a right bitwise shift. This operation effectively mimics the behavior of modulo a prime number, spreading the hashed values more uniformly across the hash table.

  Wellons delves into the mathematical underpinnings of why this method works, explaining how the multiplication with the golden ratio's fractional part and the subsequent bitwise shift are analogous to rotating a circle by an irrational angle, ensuring points are never aligned and thus promoting even distribution. He contrasts this with multiplication by a rational number, which would lead to points eventually aligning and creating clustering.

  The post further emphasizes the performance benefits of Fibonacci hashing. Since multiplication and bitwise shifts are typically faster operations than the modulo operation, especially with modern processors, Fibonacci hashing often leads to a noticeable speedup in hash table operations. This is particularly pronounced when the table size is a power of two, as the bitwise shift becomes highly optimized. The author provides some benchmark results showcasing these performance gains.

  Finally, the post acknowledges some potential drawbacks of Fibonacci hashing, such as the need for a large multiplier and the potential for bias if the initial hash function is poorly designed. However, it concludes by asserting that for the majority of use cases, Fibonacci hashing provides a superior alternative to integer modulo, especially when the hash table size is a power of two, offering improved performance and more robust hash distribution even with non-ideal hash functions. The simplicity of implementing Fibonacci hashing, requiring only multiplication and a bit shift, further strengthens its case as a powerful optimization technique.

  • Hashing
  • fibonacci hashing
  • optimization
  • integer modulo
  • Hash Table
  • data structures
  • algorithms
  • prime numbers
  • Golden Ratio
  • performance
  • Computer Science

  Summary of Comments ( 10 )
  https://news.ycombinator.com/item?id=43677122

  Verbose tl;dr

  HN commenters generally praise the article for clearly explaining Fibonacci hashing and its benefits over modulo. Some point out that the technique is not forgotten, being used in game development and hash table implementations within popular languages like Java. A few commenters discuss the nuances of the golden ratio's properties and its suitability for hashing, with one noting the importance of good hash functions over minor speed differences in the hashing algorithm itself. Others shared alternative hashing methods like "Multiply-with-carry" and "SplitMix64", along with links to resources on hash table performance testing. A recurring theme is that Fibonacci hashing shines with power-of-two table sizes, losing its advantages (and potentially becoming worse) with prime table sizes.

  The Hacker News post titled "Fibonacci Hashing: The Optimization That the World Forgot" (https://news.ycombinator.com/item?id=43677122) has a moderate number of comments, generating a discussion around the merits and applicability of Fibonacci hashing.

  Several commenters delve into the practicalities of Fibonacci hashing, questioning its supposed superiority over simpler modulo methods. One recurring point is the potential performance impact of multiplication on various architectures. While the article champions multiplication as faster than modulo, some commenters argue that this isn't universally true. Modern CPUs, they point out, often have efficient modulo instructions, especially when dealing with powers of two. One commenter specifically mentions that modulo by a power of two can be as simple as a bitwise AND operation, which is extremely fast. Therefore, the supposed speed advantage of Fibonacci hashing becomes less clear-cut and highly dependent on the specific hardware.

  Another key discussion thread centers around the quality of hash distribution. Some commenters express skepticism about Fibonacci hashing consistently outperforming modulo, especially when dealing with real-world data that might not be uniformly distributed. Concerns are raised about potential clustering or patterns in the hashed values that could negatively impact performance. One commenter highlights the importance of benchmarking with realistic datasets to demonstrate any tangible benefits over traditional methods. They also mention Knuth's multiplicative hashing method as a strong contender, suggesting it often provides a good balance between speed and distribution quality.

  A few commenters provide valuable context by linking to related resources and discussions. One link points to a Stack Overflow post discussing the choice of the multiplier in multiplicative hashing. Another commenter shares a link to a paper analyzing different hashing methods. These external resources add depth to the conversation and provide alternative perspectives on the topic.

  Finally, some commenters offer practical advice and considerations. One commenter suggests that the choice of hashing method should depend on the specific application and its performance requirements. They emphasize the need to profile and measure the impact of different hashing strategies rather than relying on theoretical assumptions. Another commenter points out the potential complexity of implementing Fibonacci hashing correctly, which could outweigh its theoretical benefits in some cases.

  In summary, the comments section provides a balanced perspective on Fibonacci hashing, challenging the article's claim of it being a forgotten optimization. The discussion highlights the importance of considering hardware specifics, data distribution, and practical implementation challenges when evaluating any hashing method.

• 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.