ON DICKSON’S THEOREM CONCERNING ODD PERFECT
NUMBERS
PAUL POLLACK
Abstract. A 1913 theorem of Dickson asserts that for each fixed natural
number k, there are only finitely many odd perfect numbers N with at most
k distinct prime factors. We show that the number of such N is bounded by
4
k
2
.
1. Introduction
If N is a natural number, we write σ(N) := P
d|N d for the sum of the divisors
of N. We call N perfect if σ(N) = 2N, i.e., if N is equal to the sum of its proper
divisors. The even perfect numbers were completely classified by Euclid and Euler,
but the odd perfect numbers remain utterly mysterious: despite millennia of effort,
we don’t know of a single example, but we possess no argument ruling out their
existence.
In 1913, Dickson [2] proved that for each fixed natural number k, there are only
finitely many odd perfect numbers N with ω(N) ≤ k. (Here and below, we write
ω(N) for the number of distinct prime factors of the natural number N.) The first
explicit bounds were given by Pomerance [7], who showed that any such N satisfies
N ≤ (4k)
(4k)
2
k
2
.
After the work of Heath-Brown [4], and its subsequent refinements by Cook [1] and
Nielsen [5], we know that any such N satisfies
(1) N < 2
4
k
.
In addition to an upper bound on the size of such N, it is sensible to ask for a
bound on the number of such N. The purpose of this note is to prove the following
estimate:
Theorem 1. For each positive integer k, the number of odd perfect numbers N
with ω(N) ≤ k is bounded by 4
k
2
.
It is amusing to note the typographical similarities between the bound 24
k
of (1)
and our (much smaller!) bound of 4k
2
. Theorem 1 is a corollary of the following
result that is perhaps of independent interest:
Theorem 2. Let x ≥ 1 and let k ≥ 1. The number of odd perfect N ≤ x with
ω(N) ≤ k is bounded by (log x)
k
.
The proofs are self-contained except for the use of the bound (1) and an appeal to
the following classical result of Sylvester [8]: if N is odd and perfect, then ω(N) ≥ 5.
(For a detailed account of Sylvester’s investigations into odd perfect numbers, see
[3].) Recently Nielsen [6] has shown that actually ω(N) ≥ 9.