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In particular, Mordell considered the similarity classes of n n matrices with determi- × nant 1, where two such matrices L, M are in the same class if there exists a unimodular integral matrix N such that M = N T LN. The number of such similarity classes is de- noted by hn. Then a matrix M is in the similarity class of In (the ) if and only if there exists a factorisation M = N T N with an integer matrix N. This implies that the quadratic form classically associated with the M, q(x) = xT Mx, can be written as n T T T T 2 q(x)= x Mx = x N Nx = y y = yj , (1) Xj=1 where y = Nx, and N has determinant 1. Thus, the factorisation can be used to write the quadratic form q(x) as a sum of squares of n linear factors. When n = 8, such a factorisation may not exist, as Minkowski proved in 1911 that hn [1 + n/8], so hn 2 ≥ ≥ if n = 8. Mordell showed that h8 = 2 [12], and Ko showed that h9 = 2 as well [9].

In the present paper, we revisit the question of integer matri