Faits sur n’importe quel nombre

The anecdote that gave this whole genre its shape happened in 1919. G. H. Hardy took a taxi out to the nursing home at Putney where Srinivasa Ramanujan was ill, arrived with nothing to say, and fell back on the cab's number: 1729, he offered, seemed rather a dull one. Ramanujan answered immediately that it was not dull at all — it was the smallest number expressible as a sum of two cubes in two different ways. Hardy retold the exchange more than once in print, and it has carried ever since as the clearest illustration of what he described as Ramanujan's intimacy with the integers. Hardy also quoted Littlewood's version of the same observation: that every positive integer was one of Ramanujan's personal friends. The property itself was not new. Bernard Frénicle de Bessy had written 1729 down as a sum of two cubes in two ways in 1657. What was extraordinary at Putney was the retrieval, not the discovery.

The moral of the story later hardened into folklore as the interesting number paradox. Suppose some positive integers are uninteresting. That set, being a non-empty set of positive integers, has a smallest member — and being the smallest uninteresting number is surely interesting, so it cannot belong there after all. The argument is a joke with a serious edge: it is sound about the well-ordering of the integers and useless about interestingness, because "interesting" is a vague predicate that no proof can lean on.

It has been given an empirical turn too. In 2009 Nathaniel Johnston ranked integers by how many sequences in the On-Line Encyclopedia of Integer Sequences contained them, and nominated 11630 as the smallest number then absent from every one of them — a title that later additions to the database promptly took away.

Propriétés principales

• 1729 = 7 × 13 × 19 is the smallest positive integer that is a sum of two positive cubes in two different ways: 1³ + 12³ = 9³ + 10³ = 1729.
• 6174 is Kaprekar’s constant: for any four-digit number with at least two distinct digits, repeatedly subtracting the ascending digit arrangement from the descending one — padding back to four digits with leading zeros — reaches 6174 in at most seven steps.
• 496 = 2⁴(2⁵ − 1) is the third perfect number, and like every even perfect number it is also triangular: 496 = 31 × 32 ÷ 2.
• 1089 = 33², and 1089 × 9 = 9801, which is both the reversal of 1089 and 99².
• 142857 is the repeating block of 1/7, and 142857 × 7 = 999999.
• 65537 = 2¹⁶ + 1 is the largest Fermat prime known; the only known Fermat primes are 3, 5, 17, 257 and 65537.
• For a positive integer n the digital root equals 1 + ((n − 1) mod 9), so it is 9 exactly when n is a multiple of 9.
• A non-negative integer n is triangular if and only if 8n + 1 is a perfect square — the test this page uses.

Où on les rencontre

• 65537 is the public exponent in the overwhelming majority of RSA keys in use, including TLS certificates and SSH keys: it is prime, and its binary form has only two one-bits, so an encryption takes 16 squarings and a single multiplication.
• 496 is the dimension of both gauge groups that survive anomaly cancellation in ten-dimensional superstring theory, SO(32) and E8 × E8 — Green and Schwarz, 1984.
• 42 turns up as a deliberate easter egg across software, and as the conventional default random seed in tutorials and notebooks. Both are inherited from Douglas Adams rather than from any property of the number.
• Roman numerals are still standard on clock faces, in film and television copyright dates, for monarch and pope regnal numbers, and for Super Bowls — which used Arabic numerals only once, for Super Bowl 50.
• Floor numbering routinely skips 13 in Western buildings and 4 in much of East Asia, where its name resembles the word for death. These are cultural conventions and beliefs, not mathematical properties of the numbers.
• Kaprekar’s routine and the 1089 trick are staples of school mathematics enrichment and of stage magic, because both hide a short proof behind an apparently free choice.