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arXiv · 1305.3285

On the Hermite problem for cubic irrationalities

Nadir Murru
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Abstract

In this paper, the Hermite problem has been approached finding a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In other words, the problem of writing cubic irrationals as a periodic sequence of rational or integer numbers has been solved. In particular, a periodic multidimensional continued fraction (with pre--period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences. This multidimensional continued fraction is derived from a modification of the Jacobi algorithm, which is proved periodic if and only if the inputs are cubic irrationals. Moreover, this representation provides simultaneous rational approximations for cubic irrationals.

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Nadir Murru. 2013-05-14. On the Hermite problem for cubic irrationalities. https://arxiv.org/abs/1305.3285

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