- ANCIENT GREEK NUMBER THEORY 167
When these have been discovered, 6 among the units and 28 in
the tens, you must do the same to fashion the next... the result is
496, in the hundreds; and then comes 8,128 in the thousands, and
so on, as far as it is convenient for one to follow [D'ooge, 1926, p.
211].^3This quotation seems to imply that Nicomachus expected to find one perfect number
Nk having k digits. Actually, the fifth perfect number is 33,550,336, so we have
jumped from four digits to eight here. The sixth is 8,589,869,056 (10 digits) and
the seventh is 137,438,691,328 (12 digits), so that there is no regularity about the
distribution of perfect numbers. Thus, Nicomachus was wise to refrain from making
conjectures too explicitly. According to Dickson (1919, p. 8), later mathematicians,
including Cardano, were less restrained, and this incorrect conjecture has been
stated more than once.
For a topic that is devoid of applications, perfect numbers have attracted a
great deal of attention from mathematicians. Dickson (1919) lists well over 100
mathematical papers devoted to this topic over the past few centuries. From the
point of view of pure number theory, the main questions about them are the follow-
ing: (1) Is there an odd perfect number?^4 (2) Are all even perfect numbers given
by the procedure described by Nicomachus?^5 (3) Which numbers of the form 2" -1
are prime? These are called Mersenne primes, after Marin Mersenne (1588-1648),
who, according to Dickson (1919, pp. 12-13), first noted their importance, precisely
in connection with perfect numbers. Obviously, ç must itself be prime if 2n — 1 is
to be prime, but this condition is not sufficient, since 2^11 - 1 = 23 · 89. The set
of known prime numbers is surprisingly small, considering that there are infinitely
many to choose from, and the new ones being found tend to be Mersenne primes,
mostly because that is where people are looking for them. The largest currently
known prime (as of June 2004) is 224036583 - 1, only the forty-first Mersenne prime
known.^6 It was found on May 15, 2004 by the GIMPS (Great Internet Mersenne
Prime Search) project, which links over 200,000 computers via the Internet and
runs prime-searching software in the background of each while their owners are
busy with their own work. This prime has 7,235,733 decimal digits. The fortieth
Mersenne prime, 220996011 - 1, was found on November 17, 2003; it has 6,320,430
decimal digits. In contrast, the largest known non-Mersenne prime is 3 • 2303093 -f-1,
found by Jeff Young in 1998.^7 It is rather tiny in comparison with the last few
Mersenne primes discovered, having "only" 91,241 decimal digits.
Beginning in Chapter 6 of Book 2, Nicomachus studies figurate numbers: polyg-
onal numbers through heptagonal numbers, and then polyhedral numbers. These
numbers are connected with geometry, with an identification of the number 1 with
a geometric point. To motivate this discussion Nicomachus speculated that the
(^3) D'ooge illustrates the procedure in a footnote, but states erroneously that 8191 is not a prime.
(^4) The answer is unknown at present.
(^5) The answer is yes. The result is amazingly easy to prove, but no one seems to have noticed it
until a posthumous paper of Leonhard Euler gave a proof. Victor-Amedee Lebesgue (1791-1875)
published a short proof in 1844.
(^6) The reader will correctly infer from previous footnotes that exactly 41 perfect numbers are now
known.
(^7) See his article "Large primes and Fermat factors" in Mathematics of Computation, 67 (1998),
1735-1738, which gives a method of finding probable primes of the form k · 2n + 1.