An upper-triangular 2x2 integer matrix with diagonals p and q and off-diagonal r is a difference of squares of similar matrices if and only if p and q are each differences of two squares and r meets a divisibility condition depending on p and q, with an equivalent description via congruences on p, q
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Differences of squares of upper-triangular $2\times 2$ integer matrices
An upper-triangular 2x2 integer matrix with diagonals p and q and off-diagonal r is a difference of squares of similar matrices if and only if p and q are each differences of two squares and r meets a divisibility condition depending on p and q, with an equivalent description via congruences on p, q