Terry Tao explores the problem of efficiently decomposing a large factorial n! into a product of factors of roughly equal size √n. He outlines several approaches, including a naive iterative method that repeatedly divides n! by the largest integer below √n, and a more sophisticated approach leveraging prime factorization. The prime factorization method cleverly groups primes into products close to the target size, offering significant computational advantages. While both methods achieve the desired decomposition, the prime factorization technique highlights the interplay between the smooth structure of factorials (captured by their prime decomposition) and the goal of obtaining uniformly large factors. Tao emphasizes the efficiency gains from working with the prime factorization, and suggests potential generalizations and connections to other mathematical concepts like smooth numbers and the Dickman function.
Terence Tao's blog post, "Decomposing a Factorial into Large Factors," explores the fascinating intersection of combinatorics and number theory by examining how to efficiently decompose a large factorial into a product of factors, each of which exceeds a given threshold. He begins by establishing the context of the problem, highlighting its relevance to scenarios such as distributing a large number of items (represented by the factorial) into containers subject to a minimum capacity constraint. This framework sets the stage for a deeper dive into the mathematical intricacies involved.
Professor Tao then meticulously details an algorithmic approach to achieving this decomposition. The core of the algorithm revolves around a greedy strategy: iteratively extract the largest possible factor exceeding the prescribed threshold from the remaining factorial. This factor is determined by identifying the largest prime number less than or equal to the remaining factorial's square root and then computing the highest power of this prime that divides the remaining factorial. This power of the prime becomes a factor in the decomposition. This process is repeated, successively chipping away at the original factorial until the remaining portion falls below the specified threshold. The remaining portion, too small to be considered a "large factor," is then added as a final, possibly smaller, factor to complete the decomposition.
The post proceeds to demonstrate the algorithm's efficacy through a concrete example, dissecting the decomposition of 100! with a minimum factor size of 10. This illustrative example provides a tangible understanding of the algorithm's mechanics. Furthermore, Tao discusses the efficiency of this greedy approach. While he stops short of rigorously proving optimality, he provides compelling intuitive arguments suggesting that this method yields a decomposition with a near-minimal number of factors. This resonates with the principle of greedily maximizing the size of each extracted factor, thereby minimizing the number of factors required to exhaust the original factorial.
Finally, the post subtly touches upon the underlying number theory principles that underpin the algorithm's effectiveness. The process of identifying the largest prime power divisor leverages well-established number-theoretic concepts related to prime factorization and divisibility. While not explicitly delving into these theoretical underpinnings, the post subtly alludes to their importance in guaranteeing the algorithm's correctness and efficiency. In conclusion, Tao’s post provides a lucid and insightful exploration of an intriguing problem, offering a practical algorithm backed by intuitive explanations and connecting it to broader themes within number theory.
Summary of Comments ( 4 )
https://news.ycombinator.com/item?id=43506238
Hacker News users discussed the surprising difficulty of factoring large factorials, even when not seeking prime factorization. One commenter highlighted the connection to cryptography, pointing out that if factoring factorials were easy, breaking RSA would be as well. Another questioned the practical applications of this type of factorization, while others appreciated the mathematical puzzle aspect. The discussion also touched upon the computational complexity of factoring and the effectiveness of different factoring algorithms in this specific context. Some commenters shared resources and further reading on related topics in number theory. The general sentiment was one of appreciation for the mathematical curiosity presented by Terry Tao's blog post.
The Hacker News post "Decomposing a Factorial into Large Factors," linking to a blog post by Terence Tao, has generated several comments discussing the mathematics involved and its potential applications.
One commenter highlights the surprising nature of the result, expressing amazement that such a seemingly simple question about factorials could lead to a non-trivial mathematical problem. They appreciate the elegance of the proof and the unexpected connection to prime number theory.
Another commenter delves into the computational aspects, pondering the efficiency of finding such decompositions. They discuss the possibility of using known factoring algorithms or if the specific structure of the problem allows for more specialized, faster techniques. The commenter also raises the question of whether there's a practical application for this decomposition, speculating on potential uses in cryptography or coding theory.
A further comment focuses on the "smoothness" of the factors, referring to the property of having only small prime factors. They connect this to the concept of smooth numbers, which are frequently used in cryptographic algorithms, and suggest exploring the relationship further. This commenter also proposes investigating the distribution of the largest prime factor in the decomposition.
One commenter with a username related to elliptic curves briefly mentions a potential link to elliptic curve cryptography, although they don't elaborate on the specific connection. This suggests a possible avenue for further exploration, hinting at a deeper relationship between the factorial decomposition and advanced cryptographic techniques.
Another commenter points out the accessibility of Tao's explanation, praising his ability to break down complex mathematical concepts into understandable terms. They appreciate how the blog post makes the problem and its solution accessible to a wider audience.
Finally, a commenter raises a more philosophical point about the nature of mathematical discovery. They reflect on how seemingly simple questions can lead to unexpected mathematical journeys and highlight the beauty of uncovering hidden connections between different areas of mathematics. They express admiration for Tao's ability to find and illuminate these connections.
Overall, the comments on the Hacker News post demonstrate a mix of appreciation for the mathematical elegance of the result, curiosity about its computational implications, and speculation about potential applications. They also showcase the collaborative nature of the platform, where commenters build on each other's ideas and explore different facets of the topic.