Geodesic
Sums of squares over the Fibonacci $\circ$-ring
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 165-190 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper considers Diophantine equations of the form $$ X_1^2+[(X_1+1)\tau]^2+\cdots+X_k^2+[(X_k+1)\tau]^2=A, $$ where $X_i,A\in\mathbb Z$ ($A\ge 0$) are rational integers; $k=2,3,4$, $\tau=(-1+\sqrt{5})/2$ is the golden section, and $[*]$ denotes the integral part of a number. For these equations, the solvability conditions are found, and lower bounds for the number of solutions are obtained. The equations considered are closely related to equations of the form $$ X_1\circ X_1+\cdots+X_k\circ X_k=A, $$ where $\circ$ denotes the Knuth circle multiplication. Bibliography: 18 titles. 
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     title = {Sums of squares over the {Fibonacci} $\circ$-ring},
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V. G. Zhuravlev. Sums of squares over the Fibonacci $\circ$-ring. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 165-190. http://geodesic.mathdoc.fr/item/ZNSL_2006_337_a9/

[1] R. Lifshitz, “The square Fibonacci tiling”, J. Alloys and Compaunds, 342 (2002), 186–190 | DOI

[2] G. Urban, J. Wolny, “Average unit cell of a square Fibonacci tiling”, J. Non-Cryst. Solids, 2004, 334–335, 105–109

[3] D. E. Knuth, “Fibonacci multiplication”, Appl. Math. Lett., 1 (1988), 57–60 | DOI | MR | Zbl

[4] A. Messaoudi, “Tribonacci multiplication”, Appl. Math. Lett., 15 (2002), 981–985 | DOI | MR | Zbl

[5] R. V. Moody, Model sets: a survey, from Quasicrystals to more complex systems, eds. F. Alex, F. Dénoyer, J. P. Gazeau, EPD Science, Les Ulis, and Springer-Verlag, Berlin, 2000

[6] J. F. Sadoc, R. Mosseri, “The $E_8$ lattice and quasicrystals: geometry, number theory and quasicrystals”, J. Phys. A, 26 (1993), 1789–1809 | DOI | MR | Zbl

[7] R. V. Moody, J. Patera, “Quasicrystals and icosians”, J. Phys. A, 26 (1993), 2829–2853 | DOI | MR | Zbl

[8] R. V. Moody, A. Weiss, “On shelling $E_8$ quasicrystals”, J. Number Theory, 47 (1994), 405–412 | DOI | MR | Zbl

[9] A. Weiss, “On shelling icosahedral quasicrystals”, Directions in mathematical quasicrystals, GRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, 2000, 161–176 | MR | Zbl

[10] J. Morita, K. Sakamoto, “Octagonal quasicrystals and a formula for shelling”, J. Phys. A, 31 (1998), 9321–9325 | DOI | MR | Zbl

[11] M. Baake, U. Grimm, “A note on shelling”, Discrete Comput. Geom., 30 (2003), 573–589 | MR | Zbl

[12] N. P. Fogg, Substitutions in dynamics, arithmetics, and combinatorics, Lect. Notes Math., 1794, 2002 | MR | Zbl

[13] Z. I. Borevich, I. R. Shafarevich, Teoriya chisel, M., 1985 | MR

[14] J. Hardy, “Sums of two squares in a quadratic ring”, Acta Arithm., 14 (1968), 357–369 | MR | Zbl

[15] H. Cohn, G. Pall, “Sums of four squares in a quadratic ring”, Amer. J. Math., 83 (1962), 536–556 | MR

[16] E. N. Donkar, “On sums of three integral squares in algebraic number fields”, Amer. J. Math., 99 (1977), 1297–1328 | DOI | MR | Zbl

[17] H. Maass, “Über die Darstellung total positiver Zahlen des Körpers $R(\sqrt5)$ als Summe von drei Quadraten”, Abh. Math. Sem. Hamb., 14 (1941), 185–191 | DOI | MR

[18] P. Ribenboim, The book of prime number records, Springer-Verlag, Berlin, 1989 | MR