整数矩阵集上的Fermat方程

Fermat's equation in the set of integral matrices

设A是m阶可逆整数矩阵,又设S(A)={Ak|k∈Z,k≥0}。设n是正整数。文中运用矩阵特征值的性质证明了:如果A有特征值α适合|α|>21n或者n>18m2(log6m)且A的特征值都不是单位根,则方程xn+yn=zn,x,y,z∈S(A)无解(x,y,z)。

Let A be a nonsingular integral matrix of order m, and let S（A）={Aklk∈Z,k≥0}. Let n be a positive integer. In this paper, by using some properties of eigenvalues of matrices, it is proved that if either 1 A has an eigenvalue｜α｜ 〉21/nor n 〉 18m2 （log6m） and all eigenvalues of A are not roots of unity, then the equationxn＋yn=zn,x,y,z∈S（A）has no solution（x,y,z）.