I. Introduction

Dr. Carl Pomerance has shared an elegant and sobering heuristic that shows Odd Perfect Numbers are very unlikely. This heuristic was originally communicated to Joshua Zelinsky in 2003, although Dr. Pomerance apparently had similar ideas earlier. Prior to this I had seen many unsubstantiated claims that odd perfect numbers probably do not exist, but I had never seen a quantitative argument about likelyhood.

Dr. Pomerance wrote:

We know that for an odd number n for which sigma(n) = 2 mod 4, we must
have n = pm², where p is a prime with p = 1 mod 4. (We may or may
not have p dividing m, it doesn't matter in the argument.) If n is to
be perfect it must be that p | σ(m²). Thus, given a number m,
there are at most log m possibilities for p. (Actually a little less
for m large.) For such a number n = pm² we have σ(n) divisible by
p. So the "probability" that σ(n) is divisible by n is p/n =
1/m². There are at most log m possibilities for p, so the
"probability" that at least one of these works is (log m)/m². Then
sum this expression over m. Since the sum converges, it "follows" that
there are only finitely many odd perfect numbers. In fact, since we've
searched up to 10³⁰⁰ without finding any, and since m² > 10¹⁵⁰ for
an odd perfect number n > 10³⁰⁰, it may be more appropriate to sum
(log m)/m² for m > 10⁷⁵. This is < 10^-70, which is so tiny, that
it is reasonable to conjecture that there are no odd perfect numbers.

Mathematically sophisticated people will understand this heuristic immediately. For the rest of us, I've attempted to restate this with more detail in the following sections.

II. Conceptual Outline

It is well known fact that any odd perfect number must be of the form

The method is to consider an arbitrary number "m" and determine an upper bound on the "probability" that we can create a perfect number with this m. Then sum this probability up over all possible values of m to get the expected number of odd perfect numbers. Finally, conclude that the expected number is so small that it is unlikely any odd perfect numbers exist. (The number p needs no exponent because we permit p to divide m).

III. Preliminaries

The following definitions and simple results from Number Theory are needed to follow the heuristic.

1. Counting Primes

The Prime Number Theorem says the number of primes less than x is approximately x/ln(x)

2. Merten's Theorem

Many web sites discuss this, including Prime Pages and Math World

3. Primorial Asymptotic

The name "primorial" is a combination of "prime" and "factorial." It is the product of all prime numbers up to some value x, and is denoted as x#. The Prime Pages and Math World also discuss this.

Math World and this paper by Caldwell and Gallot also discuss the primorial asymptotics. The asymptotic expression ln(x#)~x is used in the heuristic.

This asymptotic expression can be derived heuristically by observing that as x increases, ln(x#) either stays the same or increases by ln(x), and the probability of increase is 1/ln(x), so the expected increase is 1.

4. Bound for Number of Distinct Prime Divisors

The smallest number with "k" distinct prime divisors is the k^th primorial. So the most distinct prime divisors for any number up to x will be the the number of divisors of the largest primorial less than or equal to x. We know from the primorial asymptotic that this is ln(x)#. And we know from the Prime Number Theorem that the number of primes less than ln(x) is ln(x)/ln(ln(x)).

5. Maximum Size of σ(m²)

σ(x) gets larger than x by having many distinct prime factors. The limiting case is when x is a primorial. We can use the Primorial Asymptotic and Merten's Theorem to bound the size of σ(m²).

IV. Count of Permitted Values of p is Less Than ln(m)

In order to make (p m²) an odd perfect number, "p" must divide σ(m²). Given a value for m, how many possible values of p are there?

σ(m²) is no more than e^γln(ln(m))m²

The number of distinct prime divisors of a number of this size is no more than

V. m² Must Divide σ(pm²)

Given a value for m and a selection for p, to be a perfect number it is necessary that m² divides σ(pm²).

The probability that m² divides a random number is 1/m². These are necessary but not sufficient conditions, so we will call numbers meeting these criteria odd perfect candidates.

VI. Total Number is Finite (Tiny)

The preceeding sections develop a probability that a particular value of "m" can be part of an odd perfect candidate. We can get the expected number of candidates by adding this value up over a range. Approximating the sum with an integral, the expected number of "m's" that can be part of an odd perfect candidate, for m > a, is

VII. Is There Any Hope?

There does seem to be one ray of hope. With the exception of 6, every even perfect number is also of the form pm² with p dividing σ(m²) and m² dividing σ(pm²). Hence this heuristic suggests no large perfect numbers of any kind, neither even nor odd. But we know of many large even perfect numbers, and we suspect they are infinite. Since the heuristic is wrong for even perfect numbers, perhaps it is also wrong for odd perfect numbers. The only point of weakness I see in the heuristic is that if you could get the probability that m² divides σ(pm²) up from 1/m² to 1/(m ln(m)), then the integral would diverge, implying an infinite number of perfect candidates.

How would you do that? You would have to build some of the factors of σ(pm²) into the definition of m, so that a smaller fraction of m² relied on random luck to factor. This is what happens for even perfect numbers. This is also the method used to build factor chains in the search of odd perfect numbers. But I don't see a way to adapt the heuristic to account for this effect.

VIII. Can the Pomerance Heuristic be Taken Seriously?

The argument to accept the Pomerance Heuristic for odd perfect numbers even though it fails for even perfect numbers is that Mersenne Primes consitiute a unique conspiracy to circumvent the randomness assumptions. Furthermore, there are precedences for special cases. Dr. Pomerance makes the argument exceptionally well when he writes:

That's the trouble with heuristic arguments. They are not proofs.
Basically the thought is to assume randomness if one doesn't
know better. With even perfects, we know better (presumably).
Many years ago, Erdös and Ulam published a heuristic argument
for Fermat's Last Theorem that worked for every exponent
starting at 4. They pointed out that their heuristic suggested
in fact that there are infinitely many counterexamples at exponent
3. This is another case where we know better.

So, with odd perfects, one would want to prove there is no grand
conspiracy to produce them, as exists with even perfects. I have no
idea how to do this, but I believe it is true, and hence believe the
heuristic argument.

Finally a related thought. I have an old paper that considers the
equation σ(n)/n = α, for a fixed rational alpha. The
variable n is assumed to have exactly k distinct prime factors.
The theorem is that there are at most finitely many solutions,
except possibly in the case that m is an odd solution to σ(m)/m = α/2,
with m having exactly k-2 distinct prime factors,
and n is an even perfect number times m. So, in this kind of
theorem, we see a big bifurcation caused by the even perfects.
This is related to the old theorem of Dickson that there are
at most finitely many odd perfect numbers with a fixed number
of distinct prime factors. This result says to me that there are no
"formulas" for odd perfects, and so it adds to the heuristic.

IX. Summary

The Pomerance Heuristic suggests there should be no large perfect numbers — neither even nore odd. Dr. Pomerance argues that Mersenne Primes are a grand conspiracy to circumvent the heuristic's randomness assumption, and that no such conspiracy exists for odd perfect numbers.

Personally, I find this argument strong enough that I intend to bias the odd perfect search in directions that will produce a higher incidence of useful side effects in the form of collectible factorizations. But I don't find the argument sufficiently overwhelming to cancel the search entirely. Advances in factoring technology have created new "low hanging fruit" that we can harvest and hope to uncover an odd conspiracy.