Numbers that are their own parts

The proper divisors of six are one, two and three, and they add to six. The proper divisors of twenty-eight are one, two, four, seven and fourteen, and they add to twenty-eight. Nothing about the definition suggests such numbers should exist at all, and the fact that they do has attracted an unreasonable quantity of attention for a very long time.

Landing exactly is the exceptional case, and it is exceptional in a lopsided way: overshooting is common, falling short is common, and hitting is so rare that only four examples were known to the Greeks and only fifty-two are known now.

The lopsidedness is worth being precise about, because it says what kind of question this is. About one number in four is abundant and the rest are deficient, so the two ordinary cases have positive densities; the perfect ones have density zero and are not merely uncommon but vanishingly so. A quantity that jumps around a target and hits it only for numbers of one very particular shape is the sort of thing that has either a complete answer or none, and this question has both — a complete answer for even numbers, and after two thousand years nothing at all for odd ones.

It is also worth noticing what the definition is not. It is not a statement about the number’s size, its digits, or its position among the primes. It is a statement about the divisor lattice — that the sum over the box, minus its top corner, returns the corner — and everything below is that one equation being pushed at.

Euclid’s construction #

Book IX of the Elements ends with a proposition that looks out of place among the geometry: if $1 + 2 + 4 + \cdots + 2^{k-1}$ is prime, then that sum multiplied by the last term is perfect.

In modern notation, if $2^k - 1$ is prime then $2^{k-1}(2^k - 1)$ is perfect. For $k = 2$ that gives $6$; for $k = 3$, $28$; for $k = 5$, $496$; for $k = 7$, $8128$.

The rectangle is the proof. The divisors of $2^{k-1}q$, with $q = 2^k - 1$ prime, are exactly the powers of two up to $2^{k-1}$ in one row and those powers times $q$ in the other — there is nothing else, because $q$ is prime and contributes only itself. The first row totals $2^k - 1 = q$; the second totals $q^2$; the whole rectangle is $q + q^2 = q(1 + q) = q \cdot 2^k$, which is exactly twice $2^{k-1}q$.

Adding to twice the number is the same as the proper divisors adding to the number, since the full list includes the number itself. So perfection is the statement $\sigma(n) = 2n$, and the rectangle makes it a fact about two row totals.

Everything depends on a prime of one shape #

Euclid’s construction needs $2^k - 1$ to be prime. Numbers of that form are the Mersennes, and they are prime much less often than a first look suggests.

$2^{11} - 1 = 2047 = 23 \times 89$, so $k = 11$ produces nothing despite $11$ being prime. That is the first failure and it was found late — Mersenne’s own list of 1644 included errors that took until 1947 to fully resolve.

Two things are easy to prove. If $k$ is composite then $2^k - 1$ is composite, because $2^{ab} - 1$ is divisible by $2^a - 1$; so only prime exponents are worth testing. And that is the only general constraint anybody has: which prime exponents work is decided case by case, by computation.

The consequence is that the whole subject of even perfect numbers is a subject about Mersenne primes wearing different clothes. Each Mersenne prime gives one perfect number and each even perfect number comes from one — so the two lists are the same list, and the fifty-two known perfect numbers are the fifty-two known Mersenne primes with a factor attached.

Whether that list is finite is unknown. There are infinitely many primes, and the elementary arguments that establish it say nothing about primes of a particular shape; the heuristics suggest infinitely many Mersennes and heuristics are not proofs. It is one of the oldest open questions in the subject and it has the awkward property of being extremely easy to state and completely resistant.

Euler’s converse #

Euclid showed that his construction produces perfect numbers. Two thousand years later Euler showed it produces all the even ones, and the proof fits in a paragraph.

Let $n$ be even and perfect, and write $n = 2^{k-1}m$ with $m$ odd and $k \ge 2$. Since $\sigma$ is multiplicative and $2^{k-1}$ and $m$ are coprime,

$$\sigma(n) = \sigma(2^{k-1})\,\sigma(m) = (2^k - 1)\,\sigma(m),$$

and perfection says this equals $2n = 2^k m$. So $(2^k - 1)\sigma(m) = 2^k m$. Since $2^k - 1$ is odd and divides $2^k m$, it divides $m$; write $m = (2^k - 1)M$. Substituting gives $\sigma(m) = 2^k M = m + M$.

Now $m$ and $M$ are both divisors of $m$, and their sum is already $\sigma(m)$, the sum of all divisors of $m$. So $m$ has exactly two divisors, meaning $M = 1$ and $m = 2^k - 1$ is prime — which is Euclid’s form.

The argument is a good example of what multiplicativity buys. Nothing was factored, nothing was searched, and the entire structure of the even perfect numbers falls out of splitting $\sigma$ across a coprime pair and following the arithmetic.

How the search is actually done #

Testing whether $2^k - 1$ is prime by any general method is hopeless at the sizes involved, and the search continues anyway because Mersenne numbers have a test of their own.

The Lucas–Lehmer test takes the sequence $s_0 = 4$, $s_{i+1} = s_i^2 - 2$, computed modulo $2^k - 1$, and declares $2^k - 1$ prime exactly when $s_{k-2} = 0$. That is $k - 2$ squarings, each modulo a number of $k$ bits, and the modular reduction is nearly free because reducing modulo $2^k - 1$ is a shift and an add. A test that would be impossible in general becomes a few million operations.

What makes this worth a paragraph is the shape of the situation rather than the algorithm. The record for the largest known prime is held by a Mersenne not because such numbers are commoner but because they are the only ones anybody can certify at that size, and the same is true of perfect numbers by Euler’s theorem. Two thousand years of interest in a divisor-sum property has ended up as a special-purpose primality test running on volunteers’ computers, and every new perfect number arrives as a by-product of a prime hunt.

The odd case, and what is known about a thing nobody has found #

No odd perfect number is known. None has been proved impossible. What exists instead is a remarkable body of knowledge about the properties such a number would have to have.

An odd perfect number must be larger than $10^{1500}$. It must have at least ten distinct prime factors, and at least $101$ prime factors counted with multiplicity. Its largest prime factor must exceed $10^8$, its second largest $10^4$, its third $100$. It must be of the form $p^a m^2$ with $p \equiv a \equiv 1 \pmod 4$ — a constraint Euler proved, and which is a parity argument on $\sigma$. It cannot be divisible by $105$. It is not a perfect square.

Each of those is a theorem about an object that may not exist, and together they are one of the more peculiar collections in mathematics: a detailed description of something nobody can produce and nobody can rule out.

The methods behind the constraint list are worth a sentence, because they are all the same method. Perfection says $\sigma(n)/n = 2$ exactly; $\sigma(n)/n$ is a product of factors $(1 + 1/p + \cdots)$, one per prime; and each factor is bounded by $p/(p-1)$, which is close to one for a large prime. So a number with few prime factors, or with only large ones, cannot reach two — the product is too small — while a number with too many small factors overshoots. Every bound in the list is that squeeze applied more carefully, and the reason the bounds keep improving is that the squeeze keeps admitting more cases to rule out.

This is progress of an unusual kind. Each new theorem makes the hypothetical number larger and stranger without bringing anybody closer to deciding whether it is there, and the constraints are now so severe that most people working on the question expect no odd perfect number exists. Expectation is not proof, and the search space is infinite in exactly the direction the bounds push.

The pattern generalises in several directions and every one of them is open. Amicable pairs, where each is the sum of the other’s divisors, are known in their hundreds of millions and it is not known whether there are infinitely many. Sociable chains of length four, five, twenty-eight exist; no chain of length three has ever been found and none has been ruled out. The aliquot sequence — iterate “sum the proper divisors” — is conjectured by Catalan to always terminate or cycle, and the fate of $276$ has been unknown since 1888.

The same rectangle, read as a triangle #

There is a second way to look at Euclid’s numbers that connects them to something with no divisors in it at all.

Every even perfect number is a triangular number. $6 = 1+2+3$, $28 = 1+2+\cdots+7$, $496 = 1+2+\cdots+31$, and in general $2^{k-1}(2^k-1)$ is the sum of the first $2^k - 1$ whole numbers, since $\tfrac12 m(m+1)$ with $m = 2^k - 1$ gives exactly that. So each one is a stack of rows as well as a rectangle of divisors.

The connection is not deep and it is a good illustration of how these identities work. Two formulas coincide because both are $\tfrac12 m(m+1)$, and a statement about adding divisors becomes a statement about adding consecutive numbers with no argument in between. Every square being a stack of odd numbers is the same kind of coincidence made visible, and the reason these keep appearing is that the small quadratic expressions have very few distinct shapes between them.

A second reading of the same form: the divisors of $2^{k-1}q$ can be added by rows, as the figures above do, or by columns, giving $(1 + q)(1 + 2 + \cdots + 2^{k-1})$ — and the equality of the two is the whole proof. Counting one collection two ways is doing the work again, as it does everywhere in this field, and the reason the theorem is Euclid’s rather than somebody’s later is that this is one of the few arguments of that kind available without algebra.

What the pictures cannot show #

Every figure here is of a number small enough to draw, and the interesting cases are not. The largest known perfect number has over eighty million digits, its rectangle has forty million columns, and nothing about it can be represented except by the formula that generates it.

More to the point, the pictures show existence — this rectangle, these row totals, this exact landing — and the open questions are all about non-existence. No drawing shows that something does not exist. The odd case cannot be pictured, because the thing to be pictured is the absence of a number satisfying constraints, and the constraints are known only as a list of properties rather than as a shape.

This is the same limitation as everywhere else in the field, in a particularly clean form: the figures prove the construction works, and every remaining question is a question about what the construction misses.

Where the interest came from #

The attention perfect numbers have received is out of proportion to their mathematical importance, and the reason is not mathematical.

Nicomachus, writing around AD 100, arranged numbers into deficient, perfect and abundant and attached moral weight to the classification, describing the perfect ones as rare and beautiful and the abundant as monstrous. Augustine wrote that God created the world in six days because six is perfect, rather than six being perfect because of the creation. The medieval commentaries on Nicomachus repeated a set of assertions — that there is exactly one perfect number in each range of powers of ten, that they alternate ending in $6$ and $8$ — all of which are false and none of which anybody checked for a thousand years.

The first is spectacularly false: there is no perfect number with five digits, and none with six, seven, eight or nine. The second is half true — they do all end in $6$ or $8$, which follows from Euclid’s form — and the alternation fails at the fifth one.

That history is worth keeping for what it demonstrates. Four data points, a suggestive pattern, and an authority repeating it produced a millennium of confident false statements about numbers anybody could have checked with a divisor list. It is the strongest available argument for the habit this collection is built on, of giving every claim a test it could fail — and the same failure, generalising confidently from the handful of cases within reach, is the one small cases lie exists to name.

Where the ladder goes next #

The tool doing the work here is $\sigma$, and its behaviour came from the divisor lattice — a rectangle whose row totals factorise. Perfection is one equation about that rectangle, and there is no reason to expect an equation picked for its aesthetic appeal to have a clean answer. It has one for even numbers and quite possibly none at all for odd ones.

The other thread runs back to the primes. Even perfect numbers exist exactly as often as Mersenne primes do, so a question about divisor sums is a question about which numbers of a special shape are prime — and that is the direction almost every elementary question in this field eventually points.