Light matrices of prime determinant
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- by Daniel Goldstein, Alfred W. Hales and Richard A. Stong PDF
- Proc. Amer. Math. Soc. 142 (2014), 805-819
Abstract:
For $A = \left (a_{i,j}\right )$ a square integer matrix of prime determinant $p$, set \[ w(A)=\sum _{i,j}\left |a_{i,j}\right |.\] We are interested in the smallest possible value $w_p$ for $w(A),$ and we show that \[ \lim _{p\rightarrow \infty } w_p/\log _2(p)=5/2.\] We also show that $w_p \leq 2.5 \log _2(p)$ if and only if $p=2,7,13,37$ or a Fermat prime. Our results can also be interpreted as being about addition chains or about presentations of finite cyclic groups.References
- R. M. Guralnick, W. M. Kantor, M. Kassabov, and A. Lubotzky, Presentations of finite simple groups: a quantitative approach, J. Amer. Math. Soc. 21 (2008), no. 3, 711–774. MR 2393425, DOI 10.1090/S0894-0347-08-00590-0
- Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. MR 0378456
Additional Information
- Daniel Goldstein
- Affiliation: Center for Communications Research, San Diego, California 92121-1969
- MR Author ID: 709300
- Email: dgoldste@ccrwest.org
- Alfred W. Hales
- Affiliation: Center for Communications Research, San Diego, California 92121-1969
- MR Author ID: 79950
- Email: hales@ccrwest.org
- Richard A. Stong
- Affiliation: Center for Communications Research, San Diego, California 92121-1969
- MR Author ID: 167705
- Email: stong@ccrwest.org
- Received by editor(s): April 18, 2011
- Received by editor(s) in revised form: April 11, 2012
- Published electronically: December 4, 2013
- Communicated by: Pham Huu Tiep
- © Copyright 2013 Institute for Defense Analyses
- Journal: Proc. Amer. Math. Soc. 142 (2014), 805-819
- MSC (2010): Primary 20D05; Secondary 68W30, 11Y16, 15B36
- DOI: https://doi.org/10.1090/S0002-9939-2013-11812-5
- MathSciNet review: 3148515