×

A classification of (some) Pisot-cyclotomic numbers. (English) Zbl 1084.11058

The authors call \(q\) a Pisot-cyclotomic number if it is a Pisot number such that the ring \({\mathbb Z}[q]\) is equal to the ring \({\mathbb Z}[2 \cos(2\pi/n)]\) for some positive integer \(n\). Then such a Pisot-cyclotomic number is said to have a symmetry of order \(n.\) Pisot-cyclotomic numbers are known to have applications in the study of quasicrystals and quasilattices. Although Pisot numbers exist in every real number field, the additional arithmetical condition \({\mathbb Z}[q]={\mathbb Z}[2 \cos(2\pi/n)]\) is not always satisfied, so it is not clear for which \(n\) there are Pisot-cyclotomic numbers with symmetry of order \(n.\) In this paper, the authors consider Pisot-cyclotomic numbers of degree at most \(6.\) In each case, in order to establish whether there is a particular Pisot-cyclotomic number with symmetry of order \(n\) one needs to find out whether a corresponding Diophantine equation (often Thue equation) has integer solutions or not. Among all examples given in this paper, the Pisot-cyclotomic number \(q\) whose minimal polynomial is \(x^6-30 x^5-60x^4-32x^3+3x^2+6x+1\) has the highest symmetry \(36.\) Combining computer solution of Diophantine equations with various earlier results on some specific Diophantine equations, the authors prove that their list of Pisot-cyclotomic numbers of degree 2, 3, and 4 is complete.
In conclusion, since the above mentioned degree \(6\) symmetry \(36\) example is the ”largest” known example, they ask several natural questions about Pisot-cyclotomic numbers of higher symmetry and higher degree.

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
52C23 Quasicrystals and aperiodic tilings in discrete geometry
11Y50 Computer solution of Diophantine equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[3] Burdík, Č.; Frougny, C.; Gazeau, J. P.; Krejcar, R., Beta-integers as natural counting systems for quasicrystals, J. Phys. A, 31, 30, 6449-6472 (1998) · Zbl 0941.52019
[4] Cohn, J. H.E., On square Fibonacci numbers, J. London Math. Soc., 39, 537-540 (1964) · Zbl 0127.26705
[5] Elkharrat, A.; Frougny, C.; Gazeau, J.-P.; Verger-Gaugry, J.-L., Symmetry groups for beta-lattices, Theoret. Comput. Sci., 319, 281-305 (2004) · Zbl 1068.52028
[6] Frougny, C.; Gazeau, J.-P.; Krejcar, R., Additive and multiplicative properties of point sets based on beta-integers, Theoret. Comput. Sci., 303, 2-3, 491-516 (2003), tilings of the plane · Zbl 1036.11034
[8] McDaniel, W. L.; Ribenboim, P., The square terms in Lucas sequences, J. Number Theory, 58, 1, 104-123 (1996) · Zbl 0851.11011
[9] Mignotte, M.; Pethő, A.; Roth, R., Complete solutions of a family of quartic Thue and index form equations, Math. Comp., 65, 213, 341-354 (1996) · Zbl 0853.11022
[10] Ribenboim, P., Catalan’s Conjectureare 8 and 9 the only Consecutive Powers? (1994), Academic Press Inc.: Academic Press Inc. Boston, MA · Zbl 0824.11010
[11] Thomas, E., Complete solutions to a family of cubic Diophantine equations, J. Number Theory, 34, 2, 235-250 (1990) · Zbl 0697.10011
[12] Wakabayashi, I., On a family of quartic Thue inequalities. I, J. Number Theory, 66, 1, 70-84 (1997) · Zbl 0884.11021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.