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
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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.
@article{ZNSL_2006_337_a9,
author = {V. G. Zhuravlev},
title = {Sums of squares over the {Fibonacci} $\circ$-ring},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {165--190},
publisher = {mathdoc},
volume = {337},
year = {2006},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_337_a9/}
}
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/