Vol. 10, No. 4, 2017

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Sums of squares in quaternion rings

Anna Cooke, Spencer Hamblen and Sam Whitfield

Vol. 10 (2017), No. 4, 651–664
Abstract

Lagrange’s four squares theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question for quaternion rings, focusing on squares of elements of quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infinitely many quaternion rings, and give global upper and lower bounds.

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Keywords
Waring's problem, quaternions, sums of squares
Mathematical Subject Classification 2010
Primary: 11E25, 11P05
Secondary: 11E20
Milestones
Received: 14 November 2015
Revised: 17 April 2016
Accepted: 2 May 2016
Published: 7 March 2017

Communicated by Michael E. Zieve
Authors
Anna Cooke
McDaniel College
2 College Hill
Westminster, MD 21157
United States
Spencer Hamblen
McDaniel College
2 College Hill
Westminster, MD 21157
United States
Sam Whitfield
McDaniel College
2 College Hill
Westminster, MD 21157
United States