Cab numbers, also known as Ramanujan-Hardy numbers, are positive integers that can be expressed as the sum of two positive cubes in two different ways. The smallest such number is 1729, which is 1³ + 12³ and also 9³ + 10³. The post explores these numbers, providing a formula for generating them and listing the first few examples. It delves into the mathematical underpinnings of these intriguing numbers, discussing their connection to elliptic curves and highlighting the contributions of Srinivasa Ramanujan in identifying their unique property. The author also explores a related concept: numbers expressible as the sum of two cubes in three different ways, offering formulas and examples for these less-common numerical curiosities.
This article, titled "Cab Numbers," delves into a fascinating mathematical curiosity known as the Hardy-Ramanujan number, also referred to as the "taxi-cab number," specifically 1729. The article begins by recounting the famous anecdote of the mathematician G.H. Hardy visiting his ailing colleague Srinivasa Ramanujan in a taxicab numbered 1729. Hardy remarked that the number seemed rather dull, to which Ramanujan promptly replied that it was, in fact, quite interesting, being the smallest positive integer expressible as the sum of two positive cubes in two distinct ways.
The article then meticulously elaborates on this property, demonstrating that 1729 can be expressed as both 1³ + 12³ and 9³ + 10³. It emphasizes the significance of the word "distinct," meaning that the pairs of cubes (1, 12) and (9, 10) are different, thus excluding trivial rearrangements like 12³ + 1³.
Following this initial explanation, the article expands the concept to consider the smallest numbers expressible as the sum of two cubes in three distinct ways, denoted as Taxicab(3), then four distinct ways (Taxicab(4)), and so on. It provides these values, showcasing the rapidly increasing size of these numbers as the number of required representations grows. The article also gives the generalized mathematical notation for Taxicab(n) and explains the computational challenges in finding these numbers as 'n' increases.
Furthermore, the piece explores a related concept: the smallest number that can be expressed as the sum of two positive nth powers in two different ways. It provides examples for squares, cubes (revisiting the taxicab number), fourth powers, and fifth powers, highlighting the increasing complexity of finding such numbers. It notes how the search for these generalized "taxicab" numbers becomes computationally expensive very quickly.
Finally, the article briefly touches upon the history of the investigation into these numbers, referencing Leonard Euler's work in the 18th century. It concludes by suggesting that while 1729 might appear mundane at first glance, a deeper exploration reveals its unique mathematical properties, placing it firmly within the realm of interesting numbers.
Summary of Comments ( 3 )
https://news.ycombinator.com/item?id=42782929
Hacker News users discuss the surprising mathematical properties of "cab numbers" (integers expressible as the sum of two positive cubes in two different ways), focusing on Ramanujan's famous encounter with the number 1729. Several commenters delve into the history and related mathematical concepts, including taxicab numbers of higher order and the significance of 1729 in number theory. Some explore the computational aspects of finding these numbers, referencing algorithms and code examples. Others share anecdotes about Ramanujan and discuss the inherent beauty and elegance of such mathematical discoveries. A few commenters also provide links to further reading on related topics like Fermat's Last Theorem and the sum of cubes problem.
The Hacker News post titled "Cab Numbers" links to an article exploring the mathematical concept of cab numbers, which are integers expressible as the sum of two positive cubes in two different ways. The discussion in the comments section is brief, but touches on a few interesting points related to the topic.
One commenter notes the connection between cab numbers and taxicab numbers, pointing out the Ramanujan anecdote about the number 1729 being the smallest number expressible as the sum of two cubes in two different ways. This comment serves as a clarification for those unfamiliar with the mathematical history, reinforcing the main idea of the linked article.
Another commenter expresses appreciation for the simplicity and clarity of the explanation provided in the article, finding it more accessible than other resources they have encountered. This emphasizes the article's strength in presenting a complex mathematical concept in an understandable way.
A further comment expands on the mathematical properties of cab numbers, linking to a Wikipedia page about taxicab numbers, and mentioning generalized taxicab numbers (which consider sums of more than two cubes, and more ways of representation). This comment adds depth to the discussion, directing readers towards further exploration of the concept and its generalizations.
Finally, a short comment mentions the related concept of "taxi-cab" numbers, highlighting the subtle but crucial difference in the terms used and referring to the sequence listed in the Online Encyclopedia of Integer Sequences (OEIS). This contributes to the discussion by clarifying terminology and providing a readily available resource for further investigation.
While the number of comments is limited, they provide valuable context, clarification, and avenues for further exploration of the concept of cab numbers. They showcase appreciation for the linked article's clarity and contribute to a broader understanding of the topic's mathematical nuances.