Thank you jedishrfu. Yes I had heard a little about that. But to me the taxicab numbers are fascinating in their own right.jedishrfu said:Welcome to PF!
Taxicab numbers are very interesting. There was a recent historical note about them found in Ramanujan's notebooks. Apparently, he was investigating the math surrounding Fermat's Last Theorem:
## a^n + b^n = c^n##
and was collecting them and near misses to study it in the cubic case.
https://phys.org/news/2015-10-mathematicians-magic-key-ramanujan-taxi-cab.html
Did you follow up the references linked on the OEIS page (some of them are paywalled but at least one is free)? By 'follow up' I mean read the linked papers and make sure you understand the relevant parts, then read relevant papers referenced in those papers etc.Larry Lacey said:Yes pbuk, a very good resource on providing the sequence of taxicab numbers but not on the "structure" behind the sequences.
Taxicab numbers are positive integers that can be expressed as the sum of two positive cubes in more than one way. They are named after the famous anecdote where mathematician G.H. Hardy took a taxicab with the number 1729 and claimed it to be a dull number, but later realized it was the smallest number that can be expressed as the sum of two cubes in two different ways.
Taxicab numbers are a topic in number theory, which is a branch of mathematics that deals with the properties and relationships of numbers. The study of taxicab numbers helps to understand the properties of numbers and their structures, as well as explore the connections between different mathematical concepts.
Yes, there are several patterns and structures that have been discovered in taxicab numbers. For example, some taxicab numbers have been found to have a specific form, such as being the sum of two cubes plus one, or being a perfect square plus one. Additionally, there are certain properties and relationships between taxicab numbers that have been observed and studied by mathematicians.
Mathematicians use a variety of methods to search for new taxicab numbers. Some use computer algorithms to generate and test potential taxicab numbers, while others rely on mathematical insights and techniques to find patterns and structures in existing taxicab numbers. Collaborative efforts and discussions among mathematicians also play a crucial role in the search for new taxicab numbers.
Taxicab numbers serve as a fascinating and challenging topic in mathematics, encouraging the exploration of number theory and the development of new mathematical techniques. They also have practical applications in fields such as cryptography and coding theory. Additionally, the study of taxicab numbers can lead to new discoveries and insights in other areas of mathematics, making them an important subject of research and discussion among mathematicians.