Lucky and unlucky numbers

Two quite separate things are called lucky numbers, and they have nothing to do with each other.

The cultural sense is mostly a story about language. In Mandarin the word for four, 四 (sì), is separated from 死 (sǐ, "death") only by tone; in Japanese one reading of 四 is shi, the same syllable as 死; in Sino-Korean both are sa. The avoidance that follows — tetraphobia — is plainly visible in building numbering across China, Japan, Korea, Taiwan and Singapore, where lifts jump from 3 to 5 or label the floor F, and hotels and hospitals omit rooms numbered 4, 14 and 24. Eight runs the other way: 八 (bā) is near enough to 發 ("to prosper"), and the Beijing Olympics opening ceremony was timed for 8:08 pm on 8 August 2008 for exactly that reason.

Europe's unlucky thirteen is harder to date than the folklore implies. The familiar explanations — the thirteenth guest at the Last Supper, Loki as the thirteenth arrival at Baldr's funeral feast — were attached to the superstition later, and there is no clear written record of thirteen being unlucky much before the seventeenth century. The backlash is better documented than the belief: Captain William Fowler founded the Thirteen Club in New York on 13 January 1882, sitting thirteen diners at table 13 at 8:13 pm, having them enter beneath a ladder; several American presidents later accepted honorary membership. Thomas W. Lawson's 1907 novel Friday, the Thirteenth, about a broker who deliberately crashes the market on that day, did much to spread the date-specific version. Italy puts its suspicion on seventeen instead, because XVII rearranges to the Latin VIXI, "I have lived" — the past tense doing the work.

The mathematical lucky numbers arrived in 1956, in a paper by V. Gardiner, R. Lazarus, N. Metropolis and S. Ulam titled "On certain sequences of integers defined by sieves". The last two are better known for the Metropolis algorithm and the Monte Carlo method. Their sieve strikes out numbers by position rather than by divisibility, and the survivors — 1, 3, 7, 9, 13, 15, 21 — thin out at close to the rate the primes do. The name is a label of convenience; nothing in the construction has anything to do with fortune.

Key properties

• None of these beliefs is a property of a number. Four is unlucky in Chinese because 四 (sì) and 死 (sǐ, "death") differ only in tone — a fact about Mandarin, not about the integer four.
• Japanese has two readings for each of the avoided digits, shi/yon for 4 and ku/kyū for 9, and it is the readings shi and ku that are homophones of 死 ("death") and 苦 ("suffering").
• XVII, seventeen in Roman numerals, is an anagram of VIXI: both use exactly the letters X, V, I and I.
• Seven is the most likely total from two fair six-sided dice — 6 of the 36 equally likely outcomes sum to 7, a probability of exactly 1/6.
• 108 = 2² × 3³. Its digits sum to 9 and it is divisible by 9, which makes it a Harshad number.
• Ulam's lucky numbers begin 1, 3, 7, 9, 13, 15, 21, 25, 31, 33 (OEIS A000959). Every term after 1 is odd, because the first pass of the sieve removes every second number.
• The two senses of "lucky number" barely overlap: 8, the most sought-after digit in Chinese numbering, cannot be a lucky number of the sieve, while 13 is one.
• Lucky numbers thin out at roughly the rate the primes do: there are 23 of them below 100 and 153 below 1,000, against 25 and 168 primes.

Where they turn up

• Building numbering: lifts and floor plans in China, Hong Kong, Taiwan, Japan, Korea and Singapore routinely skip 4, 14 and 24, and North American high-rises skip 13. This is a commercial accommodation of a belief, not evidence for one.
• Airline seat maps: row 13 is missing from the cabin plans of several European carriers, and Alitalia was reported to omit row 17 as well — again an accommodation of passenger discomfort rather than an engineering constraint.
• The Beijing Olympics opening ceremony was scheduled for 8:08 pm on 8 August 2008, choosing the auspicious eight deliberately and publicly.
• Ritual counting: the 108-bead japamala used for mantra repetition in Hindu and Buddhist practice, and the 108 strikes of the temple bell on New Year's Eve in Japan.
• Vehicle plates and phone numbers: numbers heavy in eights sell at a premium at official Chinese plate auctions, while plates containing 39 were reported in the international press around 2011 to be close to unsellable in Kabul.
• In number theory the lucky numbers of the sieve are studied as a foil to the primes — "lucky primes" (3, 7, 13, 31, 37, 43, …) are numbers that are both, and analogues of the twin prime and Goldbach conjectures have been posed for them.