On Shapiro’s cyclic inequality for $N=13$
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- by B. A. Troesch PDF
- Math. Comp. 45 (1985), 199-207 Request permission
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
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Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Math. Comp. 45 (1985), 199-207
- MSC: Primary 26D15; Secondary 05A20
- DOI: https://doi.org/10.1090/S0025-5718-1985-0790653-0
- MathSciNet review: 790653