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\begin{document}

\centerline{\bf SYLVESTER: USHERING IN THE MODERN ERA OF RESEARCH
ON} \centerline{\bf ODD PERFECT NUMBERS }

\vskip 20pt

\centerline{\bf Steven Gimbel} \centerline{\smallit Department of
Philosophy, Gettysburg College} \centerline{\smallit Gettysburg,
PA 17325} \centerline{\tt sgimbel@gettysburg.edu}

\vskip 10pt

\centerline{\bf John H. Jaroma} \centerline{\smallit Department of
Mathematics and Computer Science, Austin College}
\centerline{\smallit Sherman, TX 75090} \centerline{\tt
jjaroma@austincollege.edu}

\vskip 30pt

\centerline{\smallit Received: 3/16/02, Revised: 5/21/03, Accepted:
10/22/03,
Published: 10/23/03}
\vskip 30pt

\centerline{\bf Abstract} \noindent In 1888, James Joseph
Sylvester (1814-1897) published a series of papers that he hoped
would pave the way for a general proof of the nonexistence of an
odd perfect number (OPN). Seemingly unaware that more than fifty
years earlier Benjamin Peirce had proved that an odd perfect
number must have at least four distinct prime divisors, Sylvester
began his fundamental assault on the problem by establishing the
same result. Later that same year, he strengthened his conclusion
to five. These findings would help to mark the beginning of the
modern era of research on odd perfect numbers. Sylvester's bound
stood as the best demonstrated until Gradstein improved it by one
in 1925. Today, we know that the number of distinct prime divisors
that an odd perfect number can have is at least eight. This was
demonstrated by Chein in 1979 in his doctoral thesis. However, he
published nothing of it. A complete proof consisting of almost 200
manuscript pages was given independently by Hagis. An outline of
it appeared in 1980.

\par \noindent What motivated Sylvester's sudden interest in odd
perfect numbers? Moreover, we also ask what prompted this
mathematician who was primarily noted for his work in algebra to
periodically direct his attention to famous unsolved problems in
number theory? The objective of this paper is to formulate a
response to these questions, as well as to substantiate the
assertion that much of the modern work done on the subject of odd
perfect numbers has as it roots, the series of papers produced by
Sylvester in 1888.

\pagestyle{myheadings} \markright{\smalltt INTEGERS: \smallrm
ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 3
(2003), \#A16\hfill}

\thispagestyle{empty}
\baselineskip=15pt
\vskip 30pt

%%Section 1
\noindent{\bf 1. Introduction}

\vskip 10pt

\par A perfect number, first introduced in antiquity, is any
positive integer that is equal to the sum of its proper divisors.
Euclid ($\thicksim 300$ \,B.C.) established that if $2^{p}-1$ is
prime then the resulting integer $2^{p-1}(2^{p}-1)$ is perfect.
Twenty centuries later, Euler proved that all even perfect numbers
are necessarily of Euclid's form. However, despite Euler's success
in providing a defining characteristic for all even perfect
numbers, we still do not know how many of them there are.
Furthermore, whether an odd perfect number exists or not remains
an unanswered question.

\par Having had its origin in mathematical thought more than 2000
years ago, perfect numbers did not generate a great deal of
interest until the latter part of the $19^{th}$ century. This
point is underscored in [\Ore, pg 93], when in reference to
Euler's assertion that $2^{30}(2^{31}-1)$ is perfect, Ore cites
Peter Barlow's remark that it ``is the greatest [perfect number]
that will ever be discovered, for as they are merely curious
without being useful, it is not likely that any person will
attempt to find one beyond it.'' Time, of course, proved Barlow
wrong. In 1876 Lucas demonstrated that $2^{126}(2^{127}-1)$ is
perfect. It was the tenth one to be discovered. Today, there are
thirty-nine known even perfect numbers.

\par One of the goals of this paper is to develop the argument
that the modern era of research on odd perfect numbers began with
Sylvester.\footnote{In his doctoral dissertation [\Pomerance],
Pomerance credited Sylvester with beginning the modern work on
OPNs.} To this end, we begin by stating that Euler established the
first condition of existence when he proved that if $n$ is an odd
perfect number, then

\begin{equation}
n=p^{\alpha}{q_{1}}^{2\beta_{1}}{q_{2}}^{2\beta_{2}}\cdots{q_{k}}
^{2\beta_{k}}
\end{equation}

\par\noindent where, $p, q_{1},q_{2}, \ldots, q_{k}$ are distinct
odd primes and $p\equiv\alpha\equiv 1\,($mod $4)$. However, it was
not until 1937 that Steuerwald extended this particular line of
thought by showing that not all of the $\beta_{i}$'s can equal one
[\Steuerwald].

\par Studying existence criteria from a different perspective,
Peirce provided the first known lower bound on the number of
distinct prime divisors when in 1832 he proved that an OPN must
have at least four such factors [\PeirceA]. This important result
seems to have been generally overlooked as even Dickson neglected
to mention it in his magnum opus, \emph{History of the Theory of
Numbers}, [\Dickson].

\par More than a half century after Peirce's paper appeared, both
Servais and Sylvester independently published proofs of the same
theorem (See [\Servais] and [\SylvesterI]). It is remarkable that
Sylvester makes no mention of Peirce's ground-breaking discovery
in any of his work for the two had known each other since at least
the early 1840s. This is evidenced, for example, in [\Archibald],
wherein Archibald shows a correspondence between Sylvester and
Peirce that began shortly after the former departed
Charlottesville in 1843. In [\Eliot, pg 18], it is also stated
that Sylvester was a pall-bearer at Peirce's funeral in 1880.

\par Likely, this omission of credit is attributable to
Sylvester's often demonstrated aversion for having been kept
apprised of the works of others. For instance, in [\Franklin, pg
303], Fabian Franklin, a former student and colleague of
Sylvester's at the Johns Hopkins University, described
[Sylvester's] \emph{powers} as ``being set in motion by two
opposite kinds of stimulus; that of abundantly rewarding results,
and that of the stubborn resistance of concentrated difficulty."
He further adds, ``that intermediate kind of effort which slowly
and patiently builds up and improves and perfects one's own work,
and which gives minute and prolonged study to the work of others,
he did not command in any notable degree." Parshall also notes in
[\ParshallA, pg 40] that at ``numerous times throughout his
career, Sylvester found himself in the situation of having claimed
as his own a previously published result."

\par Forthcoming, we shall describe the manner in which Sylvester's
attention was called to odd perfect numbers. But first, we present
a historical survey of the study of odd perfect numbers since the
time of Peirce.

\par In contrast to the early part of the 1800s, interest in odd
perfect numbers began to mount during the latter part of the
$19^{th}$ century with the general line of thought having mirrored
that of Peirce. In addition, up to 1888 most of the work that was
done on the subject appears to have been isolated efforts with the
dissemination of findings sometimes missing those individuals that
would turn out to be the key players. For instance, in 1863 Nocco
proved that an odd perfect number is divisible by at least three
distinct prime divisors. However, Servais, in [\ServaisA] showed
that an odd integer with either one or two prime factors cannot be
perfect. The following year, Sylvester demonstrated the three
factor case [\SylvesterG].

\par In 1888, both Sylvester and Servais independently published
proofs that an odd perfect number is divisible by at least four
different prime divisors (see [\SylvesterI] and [\Servais]). Later
that year, Sylvester advanced his bound to five [\SylvesterK]. He
furthermore showed that no odd perfect number can be divisible by
105, as well as put into place a lower bound of eight distinct
prime factors for an OPN not divisible by three [\SylvesterI].
Also in 1888, Servais, considering an odd perfect number with $k$
distinct prime divisors established an upper bound of $k+1$ on its
least prime divisor [\ServaisAA]. Improvements or extensions of
this result were later realized by Gr\"{u}n [\Grun], Cohen and
Hendy [\CohenHendy], and McDaniel [\McDaniel].

\par In 1913 Dickson showed that for any $k$ there can be only
finitely many odd perfect numbers with exactly $k$ components
[\DicksonB].\footnote{The number $n$ in Eq(1), for example, has
$k+1$ \emph{components}, namely, $p^{\alpha}$,
${q_{1}}^{2\beta_{1}}$, \ldots, ${q_{k}}^{2\beta_{k}}$.} He proved
this as a corollary to a similar result for odd primitive
non-deficient numbers (which by definition must contain all the
odd perfect numbers).\footnote{Let $n$ be any positive integer.
Then $\sigma(n)$ is the sum of the positive divisors of $n$. A
\emph{deficient number} is any positive integer $n$ such that
$\sigma(n) &lt; 2n$. Thus, $n$ is \emph{non-deficient} provided that
$\sigma(n) \geq 2n$. Dickson called a number \emph{primitive
non-deficient} provided that it is not a multiple of a smaller
non-deficient number.} Thus, one may check for the existence of an
odd perfect number with exactly $k$ components by first
delineating all of the finitely many primitive odd non-deficient
numbers associated with that k-value and then determining which
among them are equal to the sum of their proper divisors.

\par Unfortunately, this approach is not feasible for most values
of $k$. It is also questionable whether Dickson intended for his
lists of primitive non-deficient numbers to be taken too seriously
when searching for an odd perfect number. There is an extremely
large number of them, say, with up to five distinct prime factors
and they have never all been listed. Furthermore, Dickson's tables
for those with up to four distinct prime factors contain many
errors. An account of them is provided in [\CohenB].

\par In 1956, Dickson's theorem was generalized by Kanold to
include any number $n$ that satisfies $\sigma(n)/n = a/b$, where
$a, b$ are positive integers and $b \neq 0$ [\KanoldBB].

\par In 1949, Kanold [\KanoldB] produced a proof of the same
4-fold result previously given by Peirce, Servais, Sylvester, and
Dickson. However, the significance of his paper reached far beyond
the stated result. Because approaching the OPN question from the
standpoint of Dickson was considered impractical for most values
of $k$, it therefore became necessary to seek out alternative
methods for examining the possible structure of such a number. In
1974, Pomerance suggested the following class of theorems:
\emph{an OPN is divisible by j distinct primes $&gt;$ N}
[\Pomerance]. Interestingly enough, Kanold had demonstrated the
case of $j=1$ and $N=60$ using elementary techniques in 1949
[\KanoldB].

\par In 1973, by enlisting the use of computation, Hagis and
McDaniel advanced Kanold's finding to $j=1$ and $N=11200$
[\HagisMcDanielB]. Two years later, they improved $N$ to $100110$
[\HagisMcDanielC]. The first proof for $j&gt;1$ was given by
Pomerance in 1975 when he showed $j=2$ and $N=138$ [\PomeranceB].
In a paper that has recently appeared, Jenkins reports that the
largest prime divisor of an odd perfect number exceeds 10 million
[\Jenkins]. It betters the previous bound of 1 million established
by Hagis and Cohen in 1998 [\HagisC]. New estimates on the second
and third largest prime divisors of an OPN were given by Iannucci
in 1999 when he announced that they are greater than than 10000
and 100, respectively (See [\IannucciA] and [\IannucciB]).

\par The investigation of odd perfect numbers has also included
several attempts at imposing a lower bound on its overall
magnitude. In 1908, Turcaninov proved that an odd perfect number
is necessary larger than $2\cdot 10^6$. More recently, and by
integrating the use of computers, Brent, Cohen, and te Riele tell
us that it is greater than $10^{300}$ [\Brent].

\par In terms of an upper bound on its overall size, Heath-Brown
had shown in 1994 that if $n$ is an odd number with
$\sigma(n)=an$, then $n &lt; (4d)^{4^{k}}$, where $d$ is the
denominator in $a$ and $k$ is equal to the number of distinct
prime factors of $n$ [\HeathBrown]. Specifically, for an OPN this
means that $n &lt; 4^{4^{k}}$ and it sharpened the previous estimate
of $n &lt; (4k)^{(4k)^{2^{k^2}}}$ given by Pomerance in 1977
[\PomeranceC]. In reference to his own result, Heath-Brown has
noted that it is still too large to be of practical value.
Nevertheless, we point out that if it is viewed in conjunction
with the lower bound of $10^{300}$ provided by Brent et. al., then
Sylvester's theorem that every odd perfect number has at least
five distinct prime factors can be demonstrated by a
footnote.\footnote{$10^{300} &lt; n &lt; 4^{4^k}$ implies that $k &gt;
4.48$.} In 1999, Cook improved Heath-Brown's result for an OPN
with $k$ components to $n &lt; (2.124)^{4^k}$ [\Cook].
In 2003,
Nielsen refined Cook's bound to $n &lt; 2^{4^k}$ [\Nielsen]

\par A lower bound of $10^{20}$ on the largest component of an odd
perfect number was established by Cohen in 1987 [\Cohen].

\par Addressing the OPN question from a different perspective,
Steuerwald's analysis of allowable exponents continued in 1941
when Kanold discovered that it is neither possible for all of the
$\beta_{i}$'s to equal two nor for one of the $\beta_{i}$ to be
equal to two while all the rest are equal to one [\Kanold]. In
1972, Hagis and McDaniel determined that not all of the
$\beta_{i}$'s can be equal to three [\HagisMcDaniel]. In 1985,
Cohen and Williams eliminated possibilities for the $\beta_{i}$,
assuming that either some or all of the $\beta_{i}$ are the same.
They also provided a summary of all previous work done in this
area [\CohenWilliams].

\par In a recently published paper, Iannucci and Sorli restrict
the $\beta_{i}$ in order to show that an odd perfect number cannot
be divisible by three if, for all $i$, $\beta_{i} \equiv 1\,($mod
$3)$ or $\beta_{i} \equiv 2\,($mod $5)$ [\IannucciC]. In that
article, they also provided a slightly different analysis by
giving a lower bound of 37 on the total number of prime factors
(counting multiplicities) that an odd perfect number can have.

\par We now summarize some of the immediate extensions of
Sylvester's 1888 work on OPNs.

\par First, we note that unlike those that had studied the odd
perfect number question before him, Sylvester seems to have sought
to spotlight attention on both the problem itself, as well as on
his own work on it. For instance, before he proves that an OPN
cannot have less than four distinct prime divisors [\SylvesterG],
he declares that ``I am going to demonstrate that such a number
does not exist, by means of a form of reasoning with which I have
also provided a demonstration of the theorem that there does not
exist a perfect number which contains fewer than six distinct
prime factors.'' However, his proof was incorrect.

\par It was not until 1925 that Gradstein offered the first
correct demonstration of the six-component case [\Gradstein]. In
1949, and then in 1951, K\"{u}hnel [\Kuhnel] and Webber [\Webber]
independently published their own versions of this result. It
would, however, take almost fifty years years from the appearance
of Gradstein's paper for the bound of six to be improved. In 1972,
Pomerance and Robbins independently showed that an odd perfect
number has at least seven different prime divisors (see
[\Pomerance] and [\Robbins]). Several years later, in his Ph.D.
thesis [\Chein], Chein proved that an OPN has at least eight such
prime factors but he published nothing of the result. In 1980,
Hagis published an outline of a proof that consisted of almost 200
pages [\HagisA]. Hagis has also stated his belief that an
extension to nine distinct prime factors is possible but that it
would require an prohibitive amount of effort and computer time.

\par In terms of the distinct prime divisors of an odd perfect
number not divisible by three, the first improvement over
Sylvester's lower bound of eight came in 1957 when Kanold proved
that there must be at least nine such factors [\KanoldC]. The best
result known today is eleven. It was produced independently in
1983 by both Kishore [\Kishore] and Hagis [\HagisD].

\par In retrospect, we note that it was Euler who rendered the
first significant finding pertaining to the structure of an odd
perfect number. He may also very well have been the first to
comment on the caliber of the problem when in [\Euler, pg 355], he
remarked that ``whether \ldots there are any odd perfect numbers
is a most difficult question.'' More than one hundred years later,
Sylvester echoed the same sentiment in [\SylvesterG], albeit more
descriptively, when he announced that ``\ldots a prolonged
meditation on the subject has satisfied me that the existence of
any one such --- its escape, so to say, from the complex web of
conditions which hem it in on all sides --- would be little short
of a miracle. Thus then there seems to be every reason to believe
that Euclid's perfect numbers are the only perfect numbers which
exist!'' Having thus proclaimed the problem's worthiness, as well
as its ancient roots, Sylvester also declared in that paper that
he had ``found a method for determining what (if any) odd perfect
numbers exist of any specified order of manifoldness.''

\par Perhaps it was exactly this sort of exposition that Sylvester
was so often inclined to include in his work that served to create
a greater awareness of the odd perfect number problem. It may also
be possible that the wider readership of France's \emph{Comptes
Rendus}, over say, the limited purview of the \emph{New York
Mathematical Diary} had also helped to disseminate some of
Sylvester's 1888 findings on OPNs more effectively than Peirce
was able to do in 1832. In any case, Sylvester deserves the
credit for having drawn enduring attention to the problem, as
well as for having approached the problem in a manner, which with
some variations, is still manifest today.

\par While the wait continues for Sylvester's conjecture that an
odd perfect number does not exist to reach its definitive end, we
paraphrase Guy [\Guy, pg 45] when we say that perhaps it is due to
frustration over not being able to settle the existence question
of an odd perfect number that mathematicians have introduced many
similarly defined numerical concepts, as well as many problems
associated with them to study. Alas, most of them as Guy has
suggested appear to be no more tractable than the original.

\par For example, we may define a \emph{multiperfect} number to be
a positive integer $n$, where $\sigma(n)=kn$ and $k\geq1$. Such
numbers include all the perfect numbers, as well as the number $1$
(which is the only known odd example). However, a nontrivial odd
specimen or a proof of its nonexistence has yet to be found.

\par In spite of this, theory has been developed for odd
multiperfect numbers which in many ways has reflected the spirit
of approach that has come to characterize the study of OPNs. For
example, in 1906, Carmichael showed that an odd ``multiply
perfect" number has at least four distinct prime divisors
[\Carmichael]. In 1987, Kishore proved that if $n$ is odd and
$k=3$ then odd \emph{triperfect} numbers must have at least twelve
distinct prime factors [\KishoreB]. An alternative demonstration
of this result was later given by Hagis [\HagisB] with the first
proof of its kind having come in 1983 at the hand of Reidlinger
[\Reidlinger]. We remark that in 2001, a multiperfect number with
index $k=11$ was found. Although even, it was a noteworthy
discovery. As recently as early 1992 no such number with $k&gt;8$
was known.

\par We now wish to offer a few comments on \emph{amicable}
numbers, which are represented by pairs of positive integers, $n$
and $m$, satisfying $\sigma(m) = m + n = \sigma(n)$. It follows
that a number is perfect if and only if it is amicable with
itself. There are over four million known pairs of amicable
numbers [\Garcia]. However, the question, say, of their infinitude
remains an unanswered one. Furthermore, although odd pairs of
such numbers do exist (e.g. 12285 and 14595) we do not know if
there are any of opposite parity. Also open is the question of
whether or not there can be a relatively prime pair of amicable
numbers. An interesting speculation by Wall [\Wall, pg 68] asks,
``is it true that odd amicable numbers are always incongruent
modulo 4?" An affirmative answer, says Wall, would imply the
nonexistence of an odd perfect number.\footnote{This conjecture
also appears in [\Guy, pg 57]. We remark that [\Guy, pg 56] has
given te Riele's 33-digit example of an odd amicable pair not
divisible by 3. Although the illustration would appear to answer
Wall's question in the negative, we have learned that there is a
misprint. Therefore, Wall's conjecture remains alive. Our thanks
goes to Dr. Iannucci for an astute observation, as well as for
his follow-up inquiry to the author of [\Guy].}

\par Another example of a number connected by definition to a
perfect number is a \emph{quasi-perfect} number; that is, any
positive integer $n$ such that $\sigma(n)=2n+1$. Although we know
that such numbers must be odd squares, we do not know if there are
any [\Guy, pg 45].

\par On the other hand, a numerical curiosity that has proved
rather easy to find is any positive integer equal to the sum of
some of its divisors. Among other things, such numbers have been
called \emph{pseudoperfect}. However, any integer that is
abundant\footnote{$n$ is \emph{abundant} if $\sigma(n) &gt; 2n$.} but
not pseudoperfect has been labeled as \emph{weird}. We have yet to
learn if an odd weird number exists.

\par Before we conclude, we would be remiss if we did not provide
some opinion contrary to the assertion that the modern interest in
OPNs began with Sylvester. For this purpose, we cite McCarthy,
who in 1957 credited the modern revival of interest in the odd
perfect number question to Steuerwald [\McCarthy].

\par McCarthy, after having made reference to Sylvester's lower
bounds on the number of distinct prime factors of an odd perfect
number (considering both the case when the OPN is divisible by 3,
as well as when it is not), declares in [\McCarthy] that, ``after
this, progress in the [odd perfect number] problem was at a
relative standstill until recent times." Upon introducing
cyclotomic polynomials and referring the reader back to
discussions on them in secondary sources dated from 1950, he then
goes on to credit Steuerwald with initiating the modern revival of
interest in odd perfect numbers.

\par It would appear that in making his statement, McCarthy
failed to recognize certain significant achievements that occurred
during the first half of the twentieth century. Some examples
include Gradstein's 1925 direct extension of Sylvester's 1888
lower bound of five on the number of distinct prime divisors of an
OPN, as well as Dickson's 1913 theorem stating that there can be
only a finite number of odd perfect numbers having a given number
of distinct prime factors. In fact, Sylvester may have actually
anticipated the result that we now ascribe to Dickson when in
[\SylvesterG], he remarked that, ``I have found a method for
determining what (if any) odd perfect numbers exist of any
specified order of manifoldness.'' Moreover, it was Dickson's
paper that ultimately motivated attempts in the 1970s at placing
lower bounds on the magnitudes of the prime divisors of an OPN.
The first result of this kind was, in fact, realized by Kanold in
1949 prior to McCarthy having made his statement. Also not
considered was Turcaninov's 1908 lower bound on the magnitude of
an odd perfect number which has led to today's best estimate of
$10^{300}$.

\par We recognize that there are indeed some modern efforts that
reflect the ideas advanced by Steuerwald. Nevertheless, it is our
contention that the overall focus of today's present study of odd
perfect numbers is more closely related to the original work of
Sylvester, than say, to that of any other pioneer in the field.

\vskip 30pt

%%Section 2
\noindent{\bf 2. Sylvester's Motivation for Studying Odd Perfect
Numbers}

\vskip 10pt

\par We shall begin this section by stating that Sylvester's study
of odd perfect numbers appears to have been quite sudden and not
related in any obvious way to his research immediately prior.

\par We also acknowledge that the term \emph{motivation} may be a
multifaceted one insomuch as it applies to ascertaining
Sylvester's reason for embarking on the odd perfect number
problem. In one sense, motivation may suggest the historical
occurrence that has led to the endeavor; that is, the specific
event that put the question in Sylvester's mind. To this end, we
shall argue that the subject of odd perfect numbers was initially
brought to his attention by Mr. Robert W. D. Christie.

\par In a second sense, motivation may also denote the place that
Sylvester thought the problem occupied in the broader scheme of
mathematics, as well as in intellectual history as a whole. More
specifically, what did he see himself as doing if he were able to
solve this problem? In [\SylvesterG], Sylvester notes that the odd
perfect number question is an ancient problem possessing roots in
classical Greece. While Euclid's work on perfect numbers had been
known for centuries, Sylvester seems to have sought to give the
topic increased significance by also referring it back to the
works of Aristotle, Plato, and Pythagoras.

\par Thirdly, we may also interpret motivation as a manifestation
of a particular quality in Sylvester's disposition that may have
caused him to be interested in exactly this sort of a problem once
he had entertained its notion. We shall suggest that Sylvester's
interest in the question of odd perfect numbers, as well as in
other famous questions in number theory was part of a pattern he
exhibited during unsettled periods in his professional life. Three
such periods will be identified that were brought on,
respectively, by the occurrence of the following events:

\begin{center}
\begin{enumerate}
  \item An abrupt voluntary departure from the University of
  Virginia (1842)
  \item A forced retirement from the Royal Military Academy,
  Woolwich (1870)
  \item An inauspicious resignation from the Johns Hopkins University
        in order to accept the Savilian Chair of Geometry at
        Oxford University (1883)
\end{enumerate}
\end{center}

\par In order to more fully elucidate our arguments, we shall
partition our discussion into three subsections, each addressing
one of the previously stated questions of motivation.

\vskip 20pt

%%Subsection 2.1
\underline{At the Urging of Mr. Christie}

\vskip 10pt

\par The principle source for answering the first question of
motivation (i.e.,\, the event that first put the question in
Sylvester's mind), is a paper that Sylvester published in
\emph{Nature} entitled, \emph{Note on a Proposed Addition to the
Vocabulary of Ordinary Arithmetic} [\SylvesterG]. In a footnote
contained therein, Sylvester reveals that ``my particular
attention was called to perfect numbers by a letter from Mr.
Christie, dated from `Carlton Selby,' containing some inquiries
relative to the subject." Unfortunately, there were several
gentlemen with the surname of Christie whom Sylvester was likely
to have been acquainted.

\par Since the first name of Christie is not disclosed, perhaps
Sylvester felt that the identity of this individual would be
obvious to his readers. Perhaps also, he may not have considered
it a very important detail. However, the proper identification of
this person will help to produce an answer as to how and why a
specific individual would have been able to influence the research
direction of Sylvester during a professionally isolated time.

\par A difficulty that arises in making this determination is that
other than the commentaries provided by Sylvester on the subject
in his published articles, details that directly link his interest
to odd perfect numbers are scarce. A perusal of both [\Archibald]
and [\ParshallA] furthermore reveals no correspondence of
Sylvester's that specifically addresses the subject of OPNs, save
a possible allusion to them made in letter dated Feb. $26, 1888$
to Daniel Coit Gilman, President of the Johns Hopkins University
[\ParshallA, pg 269]. In particular, Sylvester writes that ``I
have not been quite idle since my accident occurred and have
recently fired off a few papers for Nature and the Comptes
Rendus." Other than this, we are aware of no other extant
communique of Sylvester that makes even the faintest reference to
an odd perfect number.

\par Nonetheless, we have been able to infer that from among the
several Christies that Sylvester had either met or likely
corresponded with, there is one that appears to have had a
demonstrated connection to number theory, perfect numbers, and
Sylvester in the vicinity of 1888. It is thus our contention that
the identity of the person to whom Sylvester refers in
[\SylvesterG] is Mr. \emph{Robert William Dougall Christie}.

\par In order to substantiate our assertion, we begin by noting
that this Christie was elected to the London Mathematical Society
in 1888, a time in which Sylvester was one of its distinguished
members. We furthermore add that this individual had been cited on
several occasions in Dickson's, \emph{History of the Theory of
Numbers}, [\Dickson] for work done on perfect numbers, as well as
in other areas of number theory. Moreover, this Christie was also
a frequent contributor to the \emph{Mathematical Questions} column
of the \emph{Educational Times and Journal of the College of
Preceptors}, a periodical that published reader's solutions to
questions posed by prominent mathematicians.\footnote{Sylvester
contributed problems to every volume of this journal from its
inception up to and including the seventieth.} In fact, in
[\ChristieB] Christie had posed the question: ``Show that the
tenth perfect number is $P_{10} = 2^{40}(2^{41}-1) =
2,417,851,639,228,158,837,784,576$." Although there is an apparent
oversight as $2^{41}-1 = 13367\cdot164511353$ is composite, this
problem nevertheless serves to establish Christie's interest in
perfect numbers around the time of 1888.\footnote{A reader, in
fact, did respond with a ``solution" to the stated problem citing
that ``The tenth perfect number is given by Mr. Carvallo in his
work `Theory of Perfect Numbers'". The reader then proceeded to
give the ``proof". We add that Dickson separately points out in
[\Dickson, pgs 22 and 24] that Carvallo had twice erroneously
announced having had a proof of the nonexistence of an odd perfect
number.}

\par An even stronger piece of evidence is found in [\ChristieA],
\emph{A Note on Perfect Numbers}, that Christie contributed to the
\emph{Mathematical Questions} in 1888. It began with, ``these
numbers [OPNs] have engaged the attention of mathematicians from
very early times, but there are still several problems connected
with them requiring a solution before the subject can be said to
be fully elucidated." Also included is an excerpt from a letter
written by Descartes to an undisclosed individual on December 20,
1638. Upon translation, it reads, ``\ldots and I don't know why
you judge that this means the invention of a true [odd] perfect
number will not be successful; if you have a demonstration, I hold
that it is beyond my capability and I would hold it in extremely
high regard; as for me, I judge that one can find real odd perfect
numbers." This citation thus provides a link between Christie and
the existence question of an \emph{odd} perfect number.

\par Perhaps the strongest evidence of all is obtained from the
paper, ``A Theorem in Combinations'', [\ChristieC], which Christie
published in the 1889 volume of the \emph{Proceedings of the
London Mathematical Society}. In particular, it concerns Euler's
$\phi$-function. This is exactly the place that Sylvester began
his discussion in [\SylvesterG] on OPNs. Moreover, excerpts from
the footnote that Sylvester included just before his mention of
Christie states, ``Euler's function $\phi(n)$, which means the
number of numbers not exceeding $n$ and prime to it, I call the
\emph{totient} of $n$ \ldots I am in the habit of representing the
totient of $n$ by the symbol $\tau{n}$, $\tau$ (taken from the
initial of the word it denotes) being a less hackneyed letter than
Euler's $\phi$, which has no claim to preference over any other
letter of the Greek alphabet, but rather the reverse." Hence, we
see Sylvester providing here an explanation of the nomenclature of
the topic of Christie's paper, immediately prior to acknowledging
him for bringing to his attention some inquiries related to the
subject of odd perfect numbers.

\vskip 20pt

%%Subsection 2.2
\underline{Sylvester's Invocation of Ancient Luminaries}

\vskip 10pt

\par The second question of Sylvester's motivation asks, ``What did
Sylvester see himself as doing if he were to solve this problem?''
That is, where exactly did Sylvester see this problem fitting into
the broader picture of number theory, or indeed, into intellectual
history as a whole? Again, we get a picture of this in
[\SylvesterG], where in the second footnote, the issue of perfect
numbers is taken up more generally.

\par After mentioning the inquiry of Christie, Sylvester then
defines a perfect number and praises Euclid's ingenuity for
showing in the ninth book of the \emph{Elements} that a number
$m$ is even perfect provided that $m=2^{n}(2^{n+1}-1)$, where
$2^{n+1}-1$ is prime. Sylvester proceeds to give a brief sketch
of Euler's proof of the converse of Euclid's theorem, upon which
he offers the comment, ``It is remarkable that Euler makes no
reference to Euclid in proving his own theorem. It must always
stand to the credit of the Greek geometers that they discovered a
class of perfect numbers which in all probability are the only
numbers which are perfect.'' Having thus duly chastised Euler for
not citing his Greek predecessors, Sylvester seeks to imbed the
problem in its full pre-Euclidean context when he adds,
``Reference is made to \emph{so-called} perfect numbers in
Plato's `Republic' H, 546B, and also by Aristotle, `Probl.' I E 3
and `Metaph.' A5.''

\par We now qualify Sylvester's references by noting that in all
cases, the term ``perfect''
[$\tau\epsilon\lambda\iota\emph{o}\upsilon$] does modify
``number'' but not in the Euclidean sense. For example, in the
cited sections of both the \emph{Problemata} [\AristotleB] and
\emph{Metaphysics} [\AristotleA], Aristotle considers what he
understands to be the claim of the number ten as ``perfect''; that
is, one which plays the role of a container for all categories of
number in the Pythagorean numerological metaphysic. Also, in the
passage from book VIII of the \emph{Republic} [\Plato], Plato
derives the so-called nuptial number. He first speaks of a cycle
that is comprehended by a ``perfect number" that governs the birth
of divine creatures and then proceeds to give a numerological
derivation of the period of this cycle for humans. Thus, in none
of the citations provided by Sylvester does it appear that the
term ``perfect" is used in its Euclidean context.

\par A possible explanation as to why Sylvester may have chosen
to include them is found in [\Franklin]. From this first-hand
account of Franklin, we learn that Sylvester had a special
interest in problems that were traceable back to the ``great
masters". In particular, Franklin recounts that ``any crucial
problem, especially one that was associated with the name of one
of the great masters, if once it attracted Sylvester's attention,
fastened itself upon his mind with a grip that seemed never to
slacken its tenacity."

\vskip 20pt

%%Subsection 2.3
\underline{Unfortunate Circumstances and Celebrated Problems in
Number Theory}

\vskip 10pt

\par Primarily noted for his work in algebra, Sylvester
occasionally demonstrated an affinity for problems in number
theory. In fact, he engaged in several such efforts during the
early 1860s while teaching at the Royal Military Academy in
Woolwich. Later, he generated a bevy of results while directing
the graduate program in mathematics at Johns Hopkins. However,
Sylvester's pursuit of longstanding, notably difficult unsolved
problems in the field appears to have been isolated to short-lived
efforts conducted during unsettled points in his career.

\par By 1888, the year in which Sylvester published his entire
collection of papers on odd perfect numbers, we find the
seventy-three year old mathematician coping with the predominantly
teaching-oriented duties of the Savilian Chair of Geometry at
Oxford. Having recently given up a satisfying position at the
newly founded Johns Hopkins University where his most recent
discoveries often became the subject of his next graduate lecture,
Sylvester arrived at Oxford in 1884 ready to teach a predominantly
undergraduate student population primarily interested in only
doing well on the university examinations. Not surprisingly, the
students were generally not very receptive to his unconventional
approach to pedagogy.

\par To provide some insight as to how Sylvester's teaching
techniques may have clashed with the undergraduate environment at
Oxford, we cite the following testimony given by E. W. Davis, a
student of Sylvester's at Hopkins: \footnote{Other examples can
also be found in [\Cajori]} ``Sylvester's Methods! He had none.
`Three lectures will be delivered on a New Universal Algebra,' he
would say; then, `The course must be extended to twelve.' It did
last all the rest of the year. The following year the course was
to be \emph{Substitutions-Theorie}, by Netto. We all got the text.
He lectured about three times, following the text closely and
stopping sharp at the end of the hour. Then he began to think
about matrices again. `I must give one lecture a week on those,'
he said. He could not confine himself to the hour, nor to the one
lecture a week. Two weeks were passed, and Netto was forgotten
entirely and never mentioned again." A historian's critique of his
classroom style is also found in [\ParshallB, pg 81] where it is
concluded that ``The Englishman was completely incapable of
satisfying the student who wished to take away a notebook
containing a crystallization of some mathematical topic."

\par Although Sylvester had proved successful from time to time
in attracting certain individuals at Oxford to take an interest in
his scholarly pursuits, the highly charged research environment
that Sylvester had helped to cultivate in Baltimore was
conspicuously absent at his new academic home. It was in part
because of this that Sylvester lamented his frustration on March
11, 1887 to Gilman and inquired about his possible return to the
Johns Hopkins [\ParshallA, pg 264]: ``\emph{Entre nous} this
University except as a school of taste and elegant light
literature is a magnificent sham. It seems to me that
Mathematical science is doomed and must eventually fall off like
a withered branch from a Tree which derives no nutriment from its
roots. \ldots I am out of heart in regard of my Professorial work
at this University in which all the real powers of influencing
the studies of the place lies in the hands of the College Tutors
\ldots The mathematical school here is at a very low ebb and the
number of Mathematical students here continually diminishing
\ldots It depends exclusively on the Tutors whether a Professor
can get undergraduates to attend his lectures \ldots Under these
circumstances I am induced to inquire from you whether you think
that there is an opening for me to return to the Johns Hopkins
\ldots I chafe under the sense of enforced inactivity at a time
when notwithstanding the weight of my years I feel in the
plentitude of my powers physical and intellectual."

\par Unfortunately, another voyage across the Atlantic never came
to pass.

\par It was under these inauspicious circumstances that the aged
Sylvester began his fundamental assault on the existence question
of an odd perfect number. This was not unlike two other earlier
periods in his life when a significant career disruption shifted
his thoughts to problems related to either Fermat's Last Theorem
or Goldbach's conjecture.

\par We shall now briefly describe the three periods of obscurity
that we alluded to earlier, for they all led to Sylvester taking
an active interest in a ``great master" problem in number theory.

\par\underline{Post-Virginia}

\par Forty-one years before his findings on odd perfect numbers
were published, Sylvester wrote three papers [\SylvesterC],
[\SylvesterD], and [\SylvesterE] related to the cubic diophantine
equation $Ax^{3} + By^{3} + Cz^{3} = Dxyz$. In the first of these
[\SylvesterC, pg 189], he remarked that ``I venture to flatter
myself that as opening out a new field in connexion with Fermat's
renowned Last Theorem, and as breaking new ground in the solution
of equations of the third degree, these results will be generally
allowed to constitute an important and substantial accession to
our knowledge of the Theory of Numbers''. This trilogy of papers
appeared at a time when Sylvester was attempting to regain his
footing as a mathematician. Due to an unpleasant and short-lived
experience at the University of Virginia, it had been several
years since he had been active in mathematics.\footnote{An
exaggerated but entertaining version of Sylvester's ordeal in
Virginia is offered by Sylvester's ex-student at Hopkins and
popularizer of mathematics, Halsted [\Halsted]. A more factual
account is, however, provided by Yates in [\Yates].}

\par In [\ParshallA, pg 3], Parshall notes that ``the years from
1842 to 1847 had been mathematically barren but by 1847, number
theory, and in particular, a problem no less formidable than
Fermat's Last Theorem, had begun to refocus his mathematical
energies." She adds on page 19, that ``in hunting for mathematical
research problems in 1847, Sylvester had big game --- Fermat's
Last Theorem --- in his sights. Significant progress on such a
famous open problem would certainly have established Sylvester
quickly as a mathematician of note."

\par By the time Sylvester had published his 1847 set of papers,
he had been employed as an actuary. Later, he met Cayley while
both were studying law. As a result, his enthusiasm for
mathematics was restored. He later re-entered academe in 1855 at
the Royal Military Academy, Woolwich.

\par The Royal Military Academy was not a research institution.
It was also considered to be inferior to its French counterpart,
\emph{l'\`{E}cole Polytechnique}. In addition, Sylvester would
occasionally engage in squabbles with the military governorship
over matters related to his teaching duties. Nevertheless, he
subsisted and remained relatively comfortable there for fifteen
years. By 1870, a change in military regulations forced him into
retirement at the age of fifty-five. The hiatus would prove to be
temporary.

\par\underline{Post-Woolwich}

\par After a nearly one-year ordeal to secure a pension, Sylvester
devoted a considerable amount of time and energy to his avocations
of poetry and singing. His first book-length study on
versification, \emph{The Laws of Verse} [\SylvesterN], was
published in 1871. It proved to be a source of great pride for
him. He followed this up with a private printing of
\emph{Fliegende Bl\"{a}tter, Supplement to the Laws of Verse},
[\SylvesterO]. During this period of professional repose, his
mathematical output would consist of eight short articles. Among
them, however, was [\SylvesterF]. It addressed the famous
conjecture of Goldbach.

\par Sylvester also provided a short discussion of this famous
problem in \emph{The Laws of Verse} [\SylvesterN, pg 123]. He
admitted that it was not due to ``Euler's correspondence with
Goldbach'' that caused him to become aware of the problem's
existence, but rather, Sylvester claims to have ``re-discovered''
the problem in connection with a theory of his regarding cubic
forms. This claimed accidental discovery also bears some
resemblance to his later account of how he first encountered the
odd perfect number question.

\par Regardless of the exact manner of his introduction to
Goldbach's conjecture, Sylvester chose not to spend a great deal
of time on trying to prove it. Rather, he opted to describe a plan
that he hoped would ultimately lead to a demonstration that the
probability of the conjecture being true can be made to be as
close to unity as one pleases.

\par By the fall of 1875, Gilman, who had been appointed president
of the newly founded Johns Hopkins University arrived in London on
a faculty finding expedition. His objective was to recruit
internationally renowned scholars to help bring to fruition a
vision of an institution committed to teaching, research, and the
training of future researchers. Among other things, Gilman was
looking for a world-class mathematician capable of inspiring and
directing the scholarly pursuits of the mathematics graduate
students in a research-oriented university. In other words, the
position that Gilman sought to fill was perfectly suited for the
style and temperament of Sylvester.

\par Initially, Gilman was a bit skeptical of Sylvester
[\ParshallA, pg 73] for setting research credentials aside, it had
been a while since he had been active in academe. In addition,
Sylvester was not regarded as one who had honed superior teaching
skills. Moreover, the tenacity that he was capable of
demonstrating in matters of personal importance was widely known.

\par In spite of it all, perhaps the most accurate testimony of
how well Sylvester might fit into the plans of the new university
was provided in a recommendation letter written on his behalf by
Peirce [\ParshallB, pg 73]: ``If you inquire about him, you will
hear his genius universally recognized but his power of teaching
will probably said to be quite deficient. \ldots as the barn yard
fowl cannot understand the flight of the eagle, so it is the
eaglet only who will be nourished by his instruction. \ldots among
your pupils, sooner or later, there must be one, who has a genius
for geometry. He will be Sylvester's special pupil
--- the one pupil who will derive from the master, knowledge and
enthusiasm --- and that one pupil will give more reputation to
your institution than the ten thousand, who will complain of the
obscurity of Sylvester, and for whom you will provide another
class of teachers."

\par Sylvester was ultimately offered the job. However, his
steadfastness in negotiating remuneration almost caused the
appointment not to materialize.\footnote{The final agreement
called for an annual salary and housing allowance of \$5,000 and
\$1,000, respectively, with both amounts to be paid in gold.}
Sylvester arrived in Baltimore in 1876 with a renewed vigor for
mathematics and a strong desire to make good on the
responsibilities that were being entrusted to him. Moreover, he
was about to occupy the most satisfying academic appointment he
would ever hold. It therefore remains somewhat of a mystery as to
why Sylvester would decide to resign seven years later in order to
accept the Savilian Chair of Geometry at Oxford.

\par\underline{Oxford}

\par Perhaps it was the allure of a most distinguished university
in his own homeland. Perhaps it also was the prestige associated
with being the twelfth occupant of the oldest Chair (founded in
1619) of any British university and previously held by the likes
of Briggs, Wallis, and Halley. Perhaps, it may even have been a
matter of poetic justice to him that he be offered and accept
academic rank at an institution that would have certainly rejected
him years earlier on religious grounds.\footnote{Because he was
Jewish, Sylvester as a student at Cambridge refused to swear
allegiance to the thirty-nine Articles of Faith of the Church of
England. This not only precluded him from receiving the degree
that he had earned but it also denied him potential fellowships
and professorships at all Anglican institutions. In 1871, the
Universities Test Act was repealed. This made non-Anglicans, such
as Sylvester, eligible to hold positions at schools like Oxford.}
Whatever the reason may have been, an optimistic Sylvester arrived
on the grounds of Oxford in 1884.

\par At first, Sylvester found Oxford agreeable. He was able
to continue his research momentum on reciprocants (differential
invariants).\footnote{Sylvester described them as being ``a great
and unlooked for \emph{revelation} which will alter the whole face
of Analytical Geometry and also produce no less effect in the
theory of Differential Equations and Transcendental Functions."
[\ParshallA, pg 259]} He also attempted to augment his teaching
duties with graduate-level lectures geared toward the College
Tutors and Lecturers with some positive response but it did not
last long [\ParshallA, pg 238]. His attempts to institute a
learning environment similar to the one that he had enjoyed in
Baltimore had failed to materialize at Oxford.

\par By 1888, and well into his seventies, Sylvester felt
capable of producing more than the diminished expectations that
Oxford had placed on him. It was under these circumstances that
his attention redirected itself to placing a definitive answer on
the question of the existence of an odd perfect number.

\par We point out that it seems to have been either a stimulating
academic environment or the want of one that dictated the nature
of Sylvester's mathematical interests. For example, although
Woolwich in the 1850s and 1860s was not a haven for research,
Cayley was nearby. As a result, Sylvester made what some have
considered to be his chief contribution to the development of the
mathematical sciences --- Invariant Theory.\footnote{Franklin
contended in 1897 that Sylvester and Cayley's development of
invariant theory marked one of the greatest contributions of
British thought to pure mathematics since the days of Newton
[\Franklin].} Conversely, when left professionally isolated, such
as during the years that immediately followed his departures from
Virginia and Woolwich, or during his latter years at Oxford,
Sylvester was at a disadvantage. Franklin tells us in [\Franklin]
that, ``those who knew him cannot fail to be convinced that,
eminent as were his actual achievements, they do not afford a
true measure of his mathematical powers, in comparison to his
great contemporaries. For he was at once less advantageously
circumstanced than they, and in an exceptional degree subject to
the influence of his surroundings."

\par Thus far, we have tried to point to Sylvester's propensity
to consider celebrated problems in number theory during periods of
isolation and uncertainty. We shall now take this opportunity to
remark that during his career, Sylvester also cultivated an
interest in longstanding questions in other areas of mathematics.
Moreover, his work on these problems was not solely relegated to
turbulent professional times. In fact, perhaps Sylvester's single
most important achievement while at Woolwich was his proof of
Newton's Rule for locating the imaginary roots of a polynomial
equation. This was a two-hundred year old problem that had
previously eluded a rigorous demonstration by Newton himself,
Maclaurin, Waring, Euler, Lagrange, and others.

\par Before we conclude our findings, we offer the following
translation (from the French) of Sylvester's proof that an odd
perfect number contains at least five distinct prime divisors
[\SylvesterK].

\newpage

\begin{center}
\emph{\textbf{Sur L'Impossibilit\'{e} De L'Existence D'un Nombre
Parfait Impair Qui Ne Contient Pas Au Moins 5 Diviseurs Premiers
Distincts}}

\vskip 5pt

\par\noindent \emph{James Joseph Sylvester} \\
\end{center}

\par We have shown in an earlier note that an odd perfect number
with less than seven factors must be divisible by 3, but in any
case, no [odd] perfect number is divisible by 105.

\par We now add that because

\[
\frac{3}{2} \cdot \frac{11}{10} \cdot \frac{13}{12} \cdot
\frac{17}{16} = \frac{151(\frac{15}{16})}{80} &lt; 2
\]

and that in substituting 11,\, 13,\, 17 for the other elements,
one can diminish this product so that it encroaches upon either 5
or 7, thus it follows that the element 3 must be associated with 7
or with 5 in a perfect number with four elements.

\par Let us therefore suppose that such a number $N$ exists.

\par (1)\,\,\,\,\, Let 3 and 7 be two of its elements. The third
element in order of size cannot exceed 13; because
\[
\frac{3}{2} \cdot \frac{7}{6} \cdot \frac{17}{16} \cdot
\frac{19}{18} = \frac{119}{64} (1 + \frac{1}{18}) &lt; \frac{126}{64}
&lt; 2.
\]
\par ($\alpha$) \,\,\, Let 11 be the third element; because
\[
\frac{3}{2} \cdot \frac{7}{6} \cdot \frac{11}{10} \cdot
\frac{29}{28} = \frac{77}{40}(1 + \frac{1}{28}) &lt; 2,
\]
\\
one can see that the fourth element must be from among the numbers
13,\, 17,\, 19, or 23.

\par But among the elements, one must be of the form $4x + 1$.

\par Moreover, we have shown in a previous note that no perfect
number can contain the number 17 without at the same time
containing an element smaller than 67. Therefore the four elements
will be 3,\, 7,\, 11,\, 13.

\par The divisor-sum \footnote[1]{If $p$ is an element and $p^{i}$
is a component of a number $N$, we call $p^{i}$ the component of
$p$, and $\frac{p^{i+1} - 1}{p - 1}$ the divisor-sum of $p$.} of 7
cannot contain the algebraic factor $7^{9} - 1$, for else
$\frac{1}{3} \cdot \frac{7^{3} - 1}{7 - 1}, \, \frac{1}{3} \cdot
\frac{7^{9} - 1}{7^{3} - 1}$ will be divisors other than 3 and 7
of this prime sum, and further contain 13 because 13 is neither a
unilinear \footnote[2]{It is very convenient in this type of
research to use the phrase "unilinear function of $x$" to signify
$kx + 1$.} of $q$ nor a divisor of $7^{3} - 1$. Thus on this
supposition, there will be at least five distinct elements.
Therefore the divisor-sum of 7 cannot contain 9, but the component
of 3 necessarily contains $3^{2}$; consequently, because the
divisor-sum of 11 (ordinary elements and not of the form $3x + 1$)
cannot contain 3, the divisor-sum of 13 carries with it an
algebraic factor of the form $\frac{13^{3} - 1}{13 - 1}$ which is
equal to $169 + 13 + 1$. Therefore 61 will be an element greater
than 3,\, 7,\, 11,\, 13 and that is contrary to the hypothesis.

\par (1) \,\,\,\,\,($\beta$)\,\,\, Let 13 be the third element.

\par Because $\frac{3}{2} \cdot \frac{7}{6} \cdot
\frac{13}{12} \cdot \frac{23}{22} = \frac{91}{48}(1+\frac{1}{22})
&lt; 2$, the fourth element will necessarily be less than 23 and the
system of elements will be 3,\, 7,\, 13,\, 19, because 17 is
excluded.

\par The divisor-sums, either of 13 or 19, cannot contain 3;
because they necessarily contain the factors $\frac{13^{2} - 1}{13
- 1}$ and $\frac{19^{3} - 1}{19 - 1}$, and therefore $\frac{1 + 13
+ 13^{2}}{3}$, that is to say 61, and $\frac{1 + 19 + 19^{2}}{3}$,
that is to say 127.

\par Therefore the divisor-sum of 7 will algebraically contain the
factors
$\frac{1}{3}\cdot\frac{7^{9}-1}{7^{3}-1}$,\,\,$\frac{1}{3}\cdot
\frac{7^{3}-1}{7-1}$; the last is equal to 19; the first is
necessarily prime to 3,\, 7,\, 19 and, for the reason already
given, to 13.

\par We have therefore demonstrated that 7 cannot be an element of
$N$.

\par (2) \,\,\,\,\, Suppose that 3 and 5 are two of the elements.

\par \,2. \,\,\,\,\, A. \,\,\,\,\, Let 5 be the special element.

\par 2. \,\,\,\,\, A($\alpha$). \,\,\,\,\, If the index of the
element 3 is 2, then, because $1 + 3 + 3^{2} = 13$, we have the
elements $3, 5, 13$; therefore the divisor-sum of 13 will contain
3, and consequently, algebraically contain the factor
$\frac{13^{2} + 13 + 1}{3}$, that is to say, 61.

\par Hence we have the elements 3,\, 5,\, 13,\, 61.

\par But $\frac{1 + 3 + 3^{2}}{9} \cdot \frac{1 + 5}{5} \cdot
\frac{13}{12} \cdot \frac{61}{60} &lt; 2$, which is inadmissible.

\par 2. \,\,\,\,\,A($\beta$). \,\,\,\,\, We therefore suppose that
the index of the component 3 is at least 4.

\par Let 3,\, 5,\, $p$ be the three elements; the index of the
divisor-sum of $p$ cannot be 9 for then we shall have at least two
other elements greater than 3,\, 5,\, $p$ and prime to 3,\, 5,\,
$p$.

\par Let $q$ be the fourth element; the same thing will be true of
the divisor-sum of $q$.

\par Therefore the product of the divisor-sums of 3,\, 5,\, $p,\, q$
cannot contain a power of 3 greater than $3^{3}$; but it must
contain one less than $3^{4}$.

\par Thus the hypothesis that 5 is a special element is
inadmissible.

\par 2.\,\,\,\,\,B. \,\,\,\,\, Moving on to the hypothesis that 5
is an ordinary element.

\par We remark that $\frac{3}{2} \cdot \frac{5}{4} \cdot
\frac{31}{30} \cdot \frac{37}{36} &lt; 1.992 &lt; 2$.

\par Consequently, there is at least one element, call it $p$, that
does not exceed 29: I say that $p$ cannot be contained in the
divisor-sum of 5, for if that were the case, the index of this sum
would necessarily be an odd divisor in excess of some prime number
less than 31, that is to say 3,\, 5,\, 7,\, 9, or 11, of which the
last four correspond to the prime numbers 11,\, 29,\, 19, and 23.

\par It cannot be 3, for $\frac{5^{3} - 1}{5 - 1} = 31$; nor 5,
for $\frac{5^{5} - 1}{5 - 1} = 11\cdot71$ (and we have the
combination of elements 3,\, 5,\, 11,\, 71; which is inadmissible
since 5 is, by hypothesis, not special, and other elements are of
the form $4x + 3)$.

\par It cannot be 7 because it is easily shown that $5^{7} - 1$
contains neither 29 nor 9; for although it is true that (5 being a
quadratic residue of 19) $5^{9} - 1$ contains 19, it contains at
the same time $5^{3} - 1$, and we have the combination 3,\, 5,\,
19,\, 31, which is forbidden for the same reason as 3,\, 5,\,
11,\, 71.

\par We are left only with 11, but $5^{11} - 1$ does not contain
23, because 5 is not a quadratic residue of 23.

\par Hence the element 5 cannot \emph{beget} (by means of the
divisor-sum to which it responds) an element which is not outside
the limit 29.

\par The divisor-sum of such an element (if it is 11 and only if
in this case) may contain 5, but not $5^{2}$; for if it contains
$5^{2}$, we will then have at least two divisors of this prime sum
between it and 3,\, 5,\, 11.

\par We remark that the component of the special element cannot be
the power (to the exponent $4j + 1$) of a number; for, if $j &gt; 0$,
$q^{4j+2} - 1$ necessarily contains two distinct prime factors in
addition to 3,\, 5, and $p$; therefore $j = 0$; hence we see that
$q + 1$ must contain the powers of 3 and 5 contained in
$3^{2}\cdot5^{2}$, that are not contained in the divisor-sum of
the other undetermined element, which can easily be shown not to
contain 3 or 5 and not $3^{2}$, $3\cdot5$, or $5^{2}$; for on the
first or last of these hypotheses, the number of elements will be
greater than four, and on the remaining hypotheses even greater
than 5. Therefore we augment the special element to be either of
the form $2k\cdot3^{2}\cdot5 - 1$ or $2k\cdot3^{2}\cdot5^{2} - 1$:
consequently, its value cannot exceed 89; this proves nonetheless
that the $p$ of which we have spoken is not a special element.

\par Let $q$ be this element, we have

\[
q = 30\lambda - 1.
\]

\par But the divisor-sum of 5 contains neither 3 nor $p$.

\par We therefore inevitably get

\[
\frac{5^x - 1}{5 - 1} = q = 30\lambda - 1,
\]

that is to say $5^{x} - 120\lambda + 3 = 0$, which is impossible.

\par This demonstrates that hypothesis 2.\,\,\,B is inadmissible,
and finally the result is achieved that there do not exist odd
perfect numbers that are divisible by fewer than 5 prime factors;
the case of multiplicity $3, 2, 1$ has already been demonstrated
for this theorem.

\par We now add a few words on perfect numbers with five elements.

\par Here,

\[
\frac{3}{2} \cdot \frac{11}{10} \cdot \frac{13}{12}\cdot
\frac{17}{16} \cdot \frac{23}{22} &lt; 1.986,
\]

but
\[
\frac{3}{2} \cdot \frac{11}{10} \cdot \frac{13}{12}\cdot
\frac{17}{16} \cdot \frac{19}{18} &gt; 2.004.
\]

We see that for a perfect number with five elements, where 5 and 7
are missing, such elements cannot be the numerals 3,\, 11,\, 13,\,
17,\, 19.

\par But 17 (a cyclotomic number of Gauss) cannot exist without a
companion number of the form $17k \pm 1$. Therefore a perfect
number with five elements, if it exists, must necessarily have
either the elements 3,\, 5 or the elements 3,\,7.

\par I have succeeded in demonstrating each of these hypotheses;
but the proof is too long to be included here. \vskip 1pt \hskip
424pt $\square$

\vskip 15pt

\par The parallel between the last line of [\SylvesterK] and Fermat
is rather striking. Furthermore, the result to which Sylvester
makes reference does not seem to have been explicitly demonstrated
by him in any of his articles. However, we do find in [\ChristieB]
displayed prominently above Christie's erroneous question on the
perfection of $2^{40}(2^{41}-1)$, the following problem posed by
Sylvester: ``If there exist any perfect number divisible by a
prime number p of the form $2^{n}+1$, show that it must be
divisible by another prime number of the form $px \pm 1$."

\par The proof by reader, W. S. Foster, appears to be correct.

\newpage

%%Section 3
\noindent{\bf 3. Conclusion}

\vskip 10pt

\par In March of 1897, Sylvester had a paralytic seizure after
having leaned over to pick up a pen that he dropped while working
in his rooms. After that, he never spoke again. Sylvester died on
March 15 of that year. His last paper was published posthumously
in 1898 [\SylvesterQ]. It was only the second article of his to
appear after resigning from Oxford for health reasons. Both papers
were number theory efforts.

\par Sylvester was primarily known as an algebraist. In that field,
he made significant contributions to invariant theory, matrix
theory, determinant theory, and the theory of equations. Although
throughout his career he had studied problems from other areas of
mathematics, it would appear that Sylvester manifested a special
interest in notable unanswered questions in the higher arithmetic
during unsettled periods in his career.

\par In 1847, Sylvester sought to connect his trilogy of papers on
cubic diophantine equations to Fermat's Last Theorem. This marked
his first set of publications after a three-year absence from
mathematics that was brought on by an abrupt decision to leave the
University of Virginia. Shortly after being forced into retirement
from Woolwich in 1870 for being superannuated, Sylvester would
consider another problem of the same genre --- Goldbach's
conjecture. In 1888, after having made a regrettable decision to
forfeit the favorable research surroundings of the Johns Hopkins
University, Sylvester took up the existence question of an odd
perfect number while at Oxford. His intention was to prove that
such numbers do not exist.

\par Environment was not the only factor that motivated his
research in 1888. It was, as Sylvester claimed, some inquires
related to the subject and posed to him by `Mr. Christie' that
initially put the idea in his mind. Evidence seems to suggest that
the person to whom Sylvester alluded was Mr. Robert William
Dougall Christie, a number theorist and frequent contributor of
problems to the \emph{Mathematical Questions} column of the
\emph{Educational Times}.

\par Sylvester began his assault on the odd perfect number problem
in 1888 by first proving what Peirce had demonstrated more than
fifty years earlier --- that an OPN necessarily has at least four
distinct prime factors. Later that year, he extended his result to
five. He also established that an odd perfect number cannot be
divisible by 105 and showed that any such number not divisible by
three must have at least eight distinct prime factors. Unlike
Peirce, Sylvester was successful in disseminating his results to a
wide audience. In turn, this contributed to an increased level of
interest in the problem that has lasted to this day.

\par Finally, we know not whether an odd perfect number exists nor
can we be completely sure of the question's decidability. However,
should an answer someday be provided, there will exist a tall
pyramid of contributors.

\vskip 20pt

%%Section 3
\noindent{\bf Acknowledgements}
\par The authors would like to thank the following individuals for
their help on this project: $(1)$ Neville Robbins, for reading an
earlier version of this manuscript, as well as for the advice that
he generously gave. $(2)$ Douglas Iannucci, for sharing his
knowledge of and enthusiasm for odd perfect numbers, as well as
for originally posing the question of Sylvester's motivation to
the second author. $(3)$ Joan Grattan (Manuscripts and Special
Collections), Johns Hopkins University, for her help in pointing
out to us the existence of certain information in our search for
the identity of Mr. Christie. $(4)$ Two anonymous referees, whose
time, expertise and constructive criticisms indeed served to make
this a much better paper.

\vskip 30pt

\baselineskip=14 true pt
\parskip=6 true pt

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\end{enumerate}
\end{document}
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