Open Access

Erratum to: The 1729 K3 surface

Research in Number Theory20173:12

https://doi.org/10.1007/s40993-017-0076-8

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 [36] 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

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Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Emory University

References

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Copyright

© The Author(s) 2017