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

Source abstract

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 $d = -1$ d equals negative 1 with 4 ∣ n $4 \mid n$ 4 vertical bar n and d = − 3 $d = -3$ d equals negative 3 with 3 ∣ n $3 \mid n$ 3 vertical bar n , and determine the values of l $l$ l that admit solutions. We extend the analysis to k × k $k \times k$ k times k matrices, and show that for every totally real field K $K$ upper K and every even n $n$ n , all solutions are trivial. We obtain some results for odd n $n$ n , but the problem for odd n $n$ n in totally real fields of degree k ⩾ 3 $k \geqslant 3$ k greater than or slanted equals 3 remains open.