McDaniel, Wayne L. Perfect Gaussian integers. Acta Arith. 25 (1973/74), 137--144. 12A05 ----------------------------------------------------------------------------- ---- References: 0 Reference Citations: 0 Review Citations: 3 ----------------------------------------------------------------------------- ---- Let $\eta=\varepsilon\prod\pi_i{}^{k_i}$ be a Gaussian integer, $\varepsilon$ a unit, $\pi_i$ primes, $\text{Re}\,\pi_i>0$, $\text{Im}\,\pi_i\geq 0$. R. Spira [Amer. Math. Monthly 68 (1961), 120--124; MR 26 #6101] defined the divisor sum function by means of $\sigma(\eta)=\prod(1+\pi_i+\cdots+\pi_i{}^{k_i})=\prod(\pi_i{}^{k_i+1}-1)/( \pi_i-1)$. The concepts even, odd, Mersenne prime, perfect numbers were extended as follows: (i) $\eta$ is an even Gaussian integer if $(1+i)|\eta$ and an odd integer if $(1+i)\nmid\eta$. (ii) The sum $\sigma((1+i)^{k-1})=-i[(1+i)^k-1]=M_k$ is called a complex Mersenne prime if $M_k$ is a prime. (iii) $\eta$ is perfect if $\sigma(\eta)=(1+i)\eta$ and is norm-perfect if $\|\sigma(\eta)\|=2\|\eta\|$, where $\|\sigma(\eta)\|=\eta\eta^*$, the norm of $\eta$. If a (norm-) perfect number $\eta$ does not have a (norm-) perfect number as a proper divisor, then $\eta$ is primitive. The main result of the present paper is contained in the following theorem: Let $M_p$ be a complex Mersenne prime and $\varepsilon$ a unit. If $p\equiv 1 (\text{mod}\,8)$, $\eta=\varepsilon(1+i)^{p-1}M_p$ is a primitive norm-perfect number; if $p\equiv-1 (\text{mod}\,8)$, $\eta=\varepsilon(1+i)^{p-1}M_p{}^*$ is a primitive norm-perfect number. Conversely if $\eta$ is an even primitive norm-perfect number then, for some unit $\varepsilon$, either $\eta=\varepsilon(1+i)^{p-1}M_p$ and $p\equiv 1 (\text{mod}\,8)$ or $\eta=\varepsilon(1+i)^{p-1}M_p{}^*$ and $p\equiv-1 (\text{mod}\,8)$; in each case $M_p$ denotes a complex Mersenne prime. Corollary: $\eta$ is a primitive perfect number if and only if there exists a rational prime $p\equiv 1 (\text{mod}\,8)$ such that $\eta=(1+i)^{p-1}M_p$. The author gives $\eta=(1+i)^6(7+8i)^2(7+120i)$ as the simplest example of an imprimitive norm-perfect number. He also proves that if $\varepsilon$ is a unit, then (a) if $p\equiv 1 (\text{mod}\,8)$ and $M_p$ and $\sigma(M_p{}^2)$ are primes then $$\eta=\varepsilon(1+i)^{p-1}M_p{}^2(\sigma(M_p{}^2))^*$$ is an imprimitive norm-perfect number; (b) if $p\equiv-1 (\text{mod}\,8)$ and $M_p$ and $\sigma(M_p{}^{*2})$ are primes then $\eta=\varepsilon(1+i)^{p-1}M_p{}^{*2}(\sigma(M_p{}^{*2}))^*$ is an imprimitive norm-perfect number.