Divisor sums

Divisor sums carry the oldest named ideas in number theory. Book IX of Euclid's Elements, from around 300 BCE, ends with Proposition 36: if 2^p − 1 is prime, then 2^(p−1)(2^p − 1) is perfect, meaning its proper divisors add up to the number itself. That construction yields 6, 28, 496 and 8128 — the four perfect numbers known to the Greeks, and still the only four below two million. Nicomachus of Gerasa, writing around 100 CE in his Introduction to Arithmetic, supplied the vocabulary that survives: a number is deficient, perfect or abundant according to whether its proper divisors fall short of it, equal it, or exceed it.

Euler finished Euclid's half-open result, proving that every even perfect number must have exactly Euclid's form; the proof was published only after his death. Whether an odd perfect number exists remains unknown, and is among the oldest unresolved questions in mathematics.

The sideways version — pairs in which each number's proper divisors sum to the other — grew its own tradition. The pair 220 and 284 is routinely credited to the Pythagoreans, but the attribution reaches us through much later authors and is better read as tradition than as record. Thābit ibn Qurra, working in ninth-century Baghdad, found a genuine rule for constructing such pairs; Fermat and Descartes rediscovered it independently in the 1630s, and Euler went on to produce several dozen new pairs. The small pair 1184 and 1210, which all of them had walked past, was found in 1866 by a sixteen-year-old Italian named Nicolò Paganini — no relation to the violinist.

The modern surprise came in 1984, when Guy Robin proved that the Riemann hypothesis is true if and only if σ(n) < e^γ·n·ln ln n for every n greater than 5040.

Key properties

• σ(n) adds up every divisor of n including n itself, so σ(6) = 1+2+3+6 = 12; the aliquot sum drops the n term.
• σ is multiplicative, and σ(p^a) = (p^(a+1) − 1)/(p − 1) for prime p — a geometric series.
• σ(n) = n + 1 exactly when n is prime.
• n is perfect when σ(n) = 2n. The only perfect numbers below two million are 6, 28, 496 and 8128.
• Euclid–Euler theorem: an even number is perfect if and only if it equals 2^(p−1)(2^p − 1) with 2^p − 1 prime.
• σ(n) is odd exactly when n is a perfect square or twice a perfect square.
• The Dirichlet series of σ factors as Σ σ(n)/nˢ = ζ(s)·ζ(s−1) for Re(s) > 2.
• Abundant numbers have a natural density of roughly 0.2476 — just under a quarter of all integers are abundant.

Where they turn up

• Every known perfect number is even and pairs with a Mersenne prime 2^p − 1; the search for more is carried out by the distributed GIMPS project. Whether an odd perfect number exists is open.
• Amicable pairs (220 and 284; 1184 and 1210) and sociable cycles: starting from 12496, repeatedly taking the aliquot sum gives 14288, 15472, 14536, 14264 and then returns to 12496.
• Jacobi’s four-square theorem: for odd n, the number of ways to write n as a sum of four squares — counting sign and order — is exactly 8·σ(n).
• Robin’s 1984 criterion turns the Riemann hypothesis into a concrete inequality on σ(n) for n above 5040, which is why divisor sums appear in work on the zeta function.
• Aliquot sequences and the Catalan–Dickson conjecture: whether iterating the aliquot sum always terminates or cycles is unknown, and the smallest undecided starting value is 276.
• Augustine of Hippo argued in The City of God that the six days of creation reflect 6 being a perfect number — a theological reading of the arithmetic, not a mathematical claim.