Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix $\left(f(x_i,x_i)\right)$ having f evaluated at the greatest common divisor $(x_i,x_i)$ of $x_i$ and $x_i$ as its...

The Fermat equation is solved in integral two by two matrices of determinant one as well as in finite order integral three by three matrices.