整数环上一类矩阵方程X^(n)+Y^(n)=λ^(n)I(n∈N,λ∈Z,λ≠0)的解

Solutions to a Class of Integer Matrix Equation X^(n)+Y^(n)=λ^(n)I(n∈N,λ∈Z,λ≠0)

设Z,N分别是全体整数和正整数的集合,M_(m)(Z)表示Z上m阶方阵的集合.本文运用Fermat大定理的结果证明了:对于取定的次数n∈N,n≥3,二阶矩阵方程X^(n)+Y^(n)=λ^(n)I(λ∈Z,λ≠0,X,Y∈M_(2)(Z),且X有一个特征值为有理数)只有平凡解;利用本原素因子的结果得到二阶矩阵方程X^(n)+Y^(n)=(±1)^(n)I(n∈N,n≥3,X,Y∈M_(2)(Z))有非平凡解当且仅当n=4或gcd(n,6)=1且给出了全部非平凡解;通过构造整数矩阵的方法,证明了下面的矩阵方程有无穷多组非平凡解:■n∈N,X^(n)+Y^(n)=λ^(n)I(λ∈Z,λ≠0,X,Y∈M_(n)(Z));X^(3)+Y^(3)=λ^(3)I(λ∈Z,λ≠0,m∈N,m≥2,X,Y∈M_(m)(Z)).

Let Z and N be the set of all integers and positive integers,respectively.Mm(Z)be the set of m×m matrix over Z where m∈N.In this paper,by using the result of Fermat’s Last Theorem,we show that the following second-order matrix equation has only trivial solutions:X^(n)+Y^(n)=λ^(n)I(λ∈Z,λ≠0,X,Y∈M_(2)(Z)),where X has an eigenvalue that is a rational number and n∈N,n≥3;By using the result of primitive divisors,we show that the second-order matrix equation X^(n)+Y^(n)=(±1)^(n) I(n∈N,n≥3,X,Y∈M_(2)(Z))has nontrivial solutions if and only if n=4 or gcd(n,6)=1 and all nontrivial solutions are given;By constructing integer matrix,we show that the following matrix equation has an infinite number of nontrivial solutions:■n∈N,X^(n)+Y^(n)=λ^(n)I(λ∈Z,λ≠0,X,Y∈M_(n)(Z));X^(3)+Y^(3)=λ^(3)I(λ∈Z,λ≠0,m∈N,m≥2,X,Y∈M_(m)(Z)).