McDaniel, Wayne L.
Perfect Gaussian integers.
Acta Arith. 25 (1973/74), 137--144.
12A05

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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.