Nombres abondants

The three-way split of the integers into abundant, perfect and deficient comes from Nicomachus of Gerasa, who set it out in his Introduction to Arithmetic around 100 CE. Nicomachus was writing a Pythagorean textbook rather than doing arithmetic for its own sake, and he presented the classification as a moral taxonomy: a number whose divisors overshoot it he described in the language of bodily excess — too many fingers, too many limbs — while deficiency was a lack and perfection the virtuous mean between them. The arithmetic held up even where the metaphor did not. The proper divisors of 12 are 1, 2, 3, 4 and 6; they total 16, so 12 is the smallest abundant number.

Boethius translated Nicomachus into Latin around 500 CE in De institutione arithmetica, and it is his vocabulary — numerus abundans, numerus deficiens — that English eventually inherited. Because that work was a set text of the medieval quadrivium, the classification was taught across Europe for roughly a thousand years, and the moral framing travelled with it.

Quantitative results came much later, once the question became statistical rather than definitional. Harold Davenport showed in 1933 that the abundancy index σ(n)/n has a continuous distribution function, which is what makes it meaningful to speak of the proportion of integers that are abundant at all. Marc Deléglise computed that proportion in 1998, pinning it between 0.2474 and 0.2480.

The modern interest in extreme abundance is sharper still. Leonidas Alaoglu and Paul Erdős named the superabundant numbers in 1944 — those more abundant than anything smaller — and in 1984 Guy Robin proved that the Riemann hypothesis is equivalent to a single inequality bounding σ(n) for every n above 5040. How far abundance can go turns out to be one of the central open questions in mathematics.

Propriétés principales

• A number is abundant when its proper divisors sum to more than itself, equivalently σ(n) > 2n. The smallest is 12, because 1 + 2 + 3 + 4 + 6 = 16.
• The smallest odd abundant number is 945 = 3³ × 5 × 7. Every odd number below 945 is deficient.
• Every multiple of an abundant number is abundant, and every proper multiple of a perfect number is abundant — so there are infinitely many.
• Abundant numbers have a natural density: Marc Deléglise showed in 1998 that it lies between 0.2474 and 0.2480, so close to one integer in four is abundant.
• Odd abundant numbers are far rarer than even ones: there are 23 below ten thousand and 1,996 below one million.
• 20161 is the largest integer that cannot be written as the sum of two abundant numbers; every larger integer can.
• An abundant number whose proper divisors are all deficient is called primitive abundant. The smallest is 20.
• No number with σ(n) = 2n + 1 — abundant by exactly one, a "quasiperfect" number — has ever been found, and whether any exists is open.

Où on les rencontre

• Project Euler Problem 23 asks for the sum of every integer that cannot be written as the sum of two abundant numbers, which makes this classification a standard exercise in writing divisor sieves.
• Robin’s criterion (Guy Robin, 1984) states that the Riemann hypothesis holds if and only if σ(n) < e^γ · n · ln ln n for every n > 5040, tying the limits of abundance to a central open problem.
• The abundancy index σ(n)/n is the main tool in the hunt for odd perfect numbers: the published lower bounds come from bounding how abundant a number with a given prime structure can be.
• Superabundant numbers, named by Alaoglu and Erdős in 1944, are those whose abundancy exceeds that of every smaller integer. They sit alongside the highly composite numbers that make 12, 60 and 360 convenient units to divide up.
• Nicomachus’s reading of abundance as excess was repeated through the medieval quadrivium and still turns up in numerology, where abundant numbers are assigned meanings the arithmetic does not support — a tradition rather than a result.