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Structured
sequences and matrix ranks
Charles Johnson, Yaoxian Qu, Duo Wang and John Wilkes
Vol. 13 (2020), No. 1, 1–8
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Abstract |
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We consider infinite sequences from a field and all matrices whose rows consist of distinct consecutive subsequences. We show that these matrices have bounded rank if and only if the sequence is a homogeneous linear recurrence; moreover, it is a -term linear recurrence if and only if the maximum rank is . This means, in particular, that the ranks of matrices from the sequence of primes are unbounded. Though not all matrices from the primes have full rank, because of the Green–Tao theorem, we conjecture that square matrices whose entries are a consecutive sequence of primes do have full rank. |
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Keywords
$k$-term linear recurrence, prime numbers, row extension,
column extension, rank, matrix of a sequence
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Mathematical Subject Classification 2010
Primary: 15A03
Secondary: 11B25, 11B37
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Milestones
Received: 9 May 2016
Revised: 13 September 2016
Accepted: 9 April 2017
Published: 4 February 2020
Communicated by Stephan Garcia
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Authors |
| Charles Johnson | |
| Department of Mathematics The College of William & Mary Williamsburg, VA United States |
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| Yaoxian Qu | |
| Department of Mathematics University of Notre Dame South Bend, IN United States |
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| Duo Wang | |
| Department of Mathematics The College of William & Mary Williamsburg, VA United States |
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| John Wilkes | |
| Department of Mathematics The College of William & Mary Williamsburg, VA United States |
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