Vol. 13, No. 1, 2020

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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
Abstract

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 k-term linear recurrence if and only if the maximum rank is k. 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.

Keywords
$k$-term linear recurrence, prime numbers, row extension, column extension, rank, matrix of a sequence
Mathematical Subject Classification 2010
Primary: 15A03
Secondary: 11B25, 11B37
Milestones
Received: 9 May 2016
Revised: 13 September 2016
Accepted: 9 April 2017
Published: 4 February 2020

Communicated by Stephan Garcia
Authors
Charles Johnson
Department of Mathematics
The College of William & Mary
Williamsburg, VA
United States
Yaoxian Qu
Department of Mathematics
University of Notre Dame
South Bend, IN
United States
Duo Wang
Department of Mathematics
The College of William & Mary
Williamsburg, VA
United States
John Wilkes
Department of Mathematics
The College of William & Mary
Williamsburg, VA
United States