Taxicab Numbers and Their Structures: Seeking Discussions

[Larry Lacey]

TL;DR

the sequence of positive integers, which can be expressed as the sum of two positive cubes in N different ways

Hi, I'm new to PF, but was hoping that there might be people on this forum with an interest in Taxicab numbers, particular on the "structure" of such integer sequences. If yes, would be delighted to hear from you.

 

[jedishrfu]

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

 

[pbuk]

Have you tried entry A011541 in OEIS as a start point for further research?

 

[Larry Lacey]

Yes pbuk, a very good resource on providing the sequence of taxicab numbers but not on the "structure" behind the sequences. That's what I'm mainly interested in. And I may had an idea on aspects of that structure. Maybe I should find out if OEIS has a forum?

 

[Larry Lacey]

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

Thank you jedishrfu. Yes I had heard a little about that. But to me the taxicab numbers are fascinating in their own right.

 

[pbuk]

Yes pbuk, a very good resource on providing the sequence of taxicab numbers but not on the "structure" behind the sequences.

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]

Agreed pbuk. But haven't found on the OEIS site the references or refs to refs I've been looking for. But OEIS and the seqfan group might be the people to provide the info/feedback I'm seeking?