Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The validity of Shapiro’s cyclic inequality
HTML articles powered by AMS MathViewer

by B. A. Troesch PDF
Math. Comp. 53 (1989), 657-664 Request permission


A cyclic sum ${S_N}({\mathbf {x}}) = \sum {{x_i}/({x_{i + 1}} + {x_{i + 2}})}$ is formed with N components of a vector x, where in the sum ${x_{N + 1}} = {x_1}$, ${x_{N + 2}} = {x_2}$, and where all denominators are positive and all numerators are nonnegative. It is known that there exist vectors x for which ${S_N}({\mathbf {x}}) < N/2$ if $N \geq 14$ and even, and if $N \geq 24$. It has been proved that the inequality ${S_N}({\mathbf {x}}) \geq N/2$ holds for $N \leq 13$. Although it has been conjectured repeatedly that the inequality also holds for odd N between 15 and 23, this has apparently never been proved. Here we will confirm that the inequality indeed holds for all odd $N \leq 23$. This settles the question for all N.
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 26D15
  • Retrieve articles in all journals with MSC: 26D15
Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Math. Comp. 53 (1989), 657-664
  • MSC: Primary 26D15
  • DOI:
  • MathSciNet review: 983563