ROOTS OF UNITY, TOTALLY REAL FIELDS AND THE MATRIX DIOPHANTINE EQUATION X n + Y n = 2 Z n $X^n + Y^n = 2 Z^n$ upper X Superscript n Baseline plus upper Y Superscript n Baseline equals 2 upper Z Superscript n

Abstract Mallick and Mishra [‘Links between the solvability of matrix and scalar Diophantine equations’, Bull. Aust. Math. Soc. , 10.1017/S0004972726101087] observed that solutions of the matrix Diophantine equation X n + Y n = 2 Z n $X^n + Y^n = 2Z^n$ upper X Superscript n Baseline plus upper Y Superscript n Baseline equals 2 upper Z Superscript n , where X , Y , Z $X, Y, Z$ upper X comma upper Y comma upper Z are 2 × 2 $2 \\times 2$ 2 times 2 matrices in a certain matrix class G 2 ( d , l ) $G_2(d, l)$ upper G 2 left parenthesis d comma l right parenthesis , correspond to solutions of the scalar equation α n + β n = 2 γ n $\\alpha ^n + \\beta ^n = 2\\gamma ^n$ alpha Superscript n Baseline plus beta Superscript n Baseline equals 2 gamma Superscript n over Z [ d ] $\\mathbb {Z}[\\sqrt {d}]$ double struck upper Z left bracket StartRoot d EndRoot right bracket and assert that there are only trivial solutions. We show that this fails when d = − 1 $d = -1$