Vladimir I. Arnold Euler's groups of powers of prime complex integers (148K, pdf) ABSTRACT. The article describes Euler's group $\Gamma(z)$, formed by the invertible elements of the ring of the residues of the integer complex numbers modulo numbers divisible by $z$. The answers are proved for the primary cases $z=(1+i)^m$ and $z=(p+iq)^m$, $p^2+q^2=4k+3$ being prime.