# Erratum to: The 1729 *K*3 surface

- Ken Ono
^{1}and - Sarah Trebat-Leder
^{1}Email author

**Received: **31 January 2017

**Accepted: **31 January 2017

**Published: **10 February 2017

The original article was published in Research in Number Theory 2016 2:26

## 1 Erratum to: Res. Number Theory (2016) 2:26 DOI 10.1007/s40993-016-0058-2

In this version of this article that was originally published [1] the authors wish to acknowledge previous work on Ramanujan’s taxi-cab number. Several important references to previous work must be included and acknowledged for completeness. The purpose of this erratum is to point out the works which should have been mentioned in the introduction of our paper.

In his Lost Notebook (see [2]), Ramanujan offered a remarkable method for finding an infinite family of solutions to \(X^{3}+Y^{3} = Z^{3}+ W^{3}\), which involves expanding rational functions at zero and infinity. The integer 1729 was one example he produced this way. Hirschhorn later devoted four papers [3–6] to examining this. Hirschhorn proposed that Ramanujan might have used a parametrization of solutions he had previously discovered along with some recurrence relations to arrive at his rational functions. The parametrization [7, 8] is \((3\mathrm{a}^{2} + 5\mathrm{ab} - 5\mathrm{b}^{2})^{3} + (4\mathrm{a}^{2} - 4\mathrm{ab} + 6\mathrm{b}^{2})^{3} = (5\mathrm{a}^{2} - 5\mathrm{ab} - 3\mathrm{b}^{2})^{3} + (6\mathrm{a}^{2} - 4\mathrm{ab} + 4\mathrm{b}^{2})^{3}\): The results in Hirschhorn’s papers do not overlap with results obtained by the authors. They should be included in the paper for historical completeness.

## Notes

## Declarations

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Ono, K., Trebat-Leder, S.: The 1729 \(K3\) surface. Res. Number Theory
**2**, 26 (2016). doi:10.1007/s40993-016-0058-2 MathSciNetView ArticleMATHGoogle Scholar - Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part IV. Springer, New York (2013)View ArticleMATHGoogle Scholar
- Han, J.H., Hirschhorn, M.D.: Another look at an amazing identity of Ramanujan. Math. Mag.
**79**, 302–304 (2006)View ArticleMATHGoogle Scholar - Hischhorn, M.D.: An amazing identity of Ramanujan. Math. Mag.
**68**, 199–201 (1995)MathSciNetView ArticleGoogle Scholar - Hischhorn, M.D.: A proof in the spirit of Zeilberger of an amazing identity of Ramanujan. Math. Mag.
**69**, 267–269 (1996)MathSciNetView ArticleGoogle Scholar - Hischhorn, M.D.: Ramanujan and Fermat’s last theorem. Aust. Math. Soc. Gaz.
**31**, 256–257 (2004)MathSciNetGoogle Scholar - Ramanujan, S.: Question 441. J. Indian Math. Soc.
**6**, 226–227 (1914)Google Scholar - Ramanujan, S.: Question 571. J. Indian Math. Soc.
**7**, 32 (1915)Google Scholar