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.
"Sums of squares in quaternion rings." Involve 10 (4) 651 - 664, 2017. https://doi.org/10.2140/involve.2017.10.651