Nombres déficients

Deficiency is the common case, and for most of its history it was treated as the uninteresting one. Nicomachus of Gerasa introduced the idea around 100 CE in his Introduction to Arithmetic, as one of three classes into which he divided the integers according to how the sum of a number's divisors compares with the number itself. His term for this class translates as "falling short", and it belonged to a moral scheme in which abundance was excess, deficiency was lack, and perfection sat as the virtuous mean between them. Boethius rendered it numerus deficiens in De institutione arithmetica around 500 CE, and because that book was a set text of the medieval quadrivium the vocabulary survived into English largely unaltered.

What makes deficiency structurally interesting is that it is inherited downwards. Every divisor of a perfect or deficient number is itself deficient, which pulls in every prime, every power of a prime, and every factor of a perfect number. Primes are as deficient as a number can be for its size: a prime p has only the divisor 1 beneath it, so it falls short by p − 1.

The class acquired genuine open problems through aliquot sequences. Replacing a number repeatedly by the sum of its proper divisors gives a trajectory that either dies at 1, settles into a perfect or amicable or sociable cycle, or — as far as anyone can prove — might grow without limit. Eugène Catalan raised the question in 1888 and Leonard Eugene Dickson sharpened it in 1913; the Catalan–Dickson conjecture, that every such sequence terminates or becomes periodic, is still open. The smallest unresolved starting value is 276, pushed for decades without a verdict.

Powers of two mark the boundary of the class. The proper divisors of 2^k sum to 2^k − 1, falling short by exactly one, which makes every power of two "almost perfect". Whether any other almost perfect number exists remains unknown.

Propriétés principales

• A number is deficient when its proper divisors sum to less than itself, equivalently σ(n) < 2n. The first few are 1, 2, 3, 4, 5, 7, 8 and 9.
• Deficiency is the common case: roughly 75.2% of integers are deficient, the complement of the roughly 24.8% that are abundant, since perfect numbers have density zero.
• Every prime and every power of a prime is deficient. A prime p falls short by p − 1, the largest shortfall possible at that size.
• Every divisor of a perfect or deficient number is itself deficient.
• Every power of two is "almost perfect", falling short by exactly 1: the proper divisors of 256 sum to 255.
• No almost perfect number other than a power of two has ever been found, and whether one exists is an open problem.
• Every odd number below 945 is deficient. 945 = 3³ × 5 × 7 is the smallest odd number that is not.
• 1 is deficient: it has no proper divisors, so its divisor sum is 0.

Où on les rencontre

• Aliquot sequences reach 1 by passing through deficient numbers, and distributed-computing projects have been pushing the sequence that starts at 276 for decades without settling whether it terminates.
• The aliquot sum of 2^p is the Mersenne number 2^p − 1, so the search for perfect numbers is in effect a search for powers of two whose shortfall-by-one partner happens to be prime.
• Powers of two fall short by exactly one, which is the same off-by-one that gives 8-bit arithmetic its maximum: the divisors of 256 sum to 255.
• Untouchable numbers — integers that are not the divisor sum of anything — were shown by Paul Erdős in 1973 to be infinite in number, a question that descends directly from this classification.
• Boethius carried the abundant/deficient/perfect split into the medieval quadrivium, where it was standard schoolroom material in Europe for roughly a thousand years.