Perfect number on Z[i]
Edwin Clark
eclark at math.usf.edu
Fri Dec 24 20:45:44 CET 2004
On Fri, 24 Dec 2004, y.kohmoto wrote:
>
> Do Mathematicians know the idea of Gaussian Perfect number?
I don't know the details but one paper you may find of interest is
McDaniel, Wayne L.
Perfect Gaussian integers.
Acta Arith. 25 (1973/74), 137--144.
12A05
Also if you do a Google searh on all the words: Gaussian Integer Perfect
you will find a number of hits.
--Edwin
PS Here's the abstract of the McDaniel paper in case you do not have easy
access to MathSciNet:
MR0332708 (48 #11034)
McDaniel, Wayne L.
Perfect Gaussian integers.
Acta Arith. 25 (1973/74), 137--144.
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; MR0148594 (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.
Reviewed by L. Carlitz
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