Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On Shapiro's cyclic inequality for $ N=13$

Author: B. A. Troesch
Journal: Math. Comp. 45 (1985), 199-207
MSC: Primary 26D15; Secondary 05A20
MathSciNet review: 790653
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A cyclic sum $ S({\mathbf{x}}) = \Sigma \;{x_i}/({x_{i + 1}} + {x_{i + 2}})$ is formed with the N components of a vector x, where $ {x_{N + 1}} = {x_1}$, $ {x_{N + 2}} = {x_2}$, and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors x for which $ S({\mathbf{x}}) < N/2$ if $ N \geqslant 14$ and even, and if $ N \geqslant 25$. It has been proved that the inequality $ S({\mathbf{x}}) \geqslant N/2$ holds for $ N \leqslant 12$. Although it has been conjectured repeatedly that the inequality also holds for odd N between 13 and 23. this has apparently not yet been proved. Here we will confirm that the inequality indeed holds for $ N = 13$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 26D15, 05A20

Retrieve articles in all journals with MSC: 26D15, 05A20

Additional Information

Keywords: Cyclic inequality, cyclic sum, minimization
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society