Strong law of small numbers explained

In mathematics, the "strong law of small numbers" is the humorous law that proclaims, in the words of Richard K. Guy (1988):^[1] In other words, any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few. Earlier (1980) this "law" was reported by Martin Gardner.^[2] Guy's subsequent 1988 paper of the same title gives numerous examples in support of this thesis. (This paper earned him the MAA Lester R. Ford Award.)

Second strong law of small numbers

thumb|Guy gives Moser's circle problem as an example. The number of and . The first five terms for the number of regions follow a simple sequence, broken by the sixth term.

Guy also formulated a second strong law of small numbers:

Guy explains this latter law by the way of examples: he cites numerous sequences for which observing the first few members may lead to a wrong guess about the generating formula or law for the sequence. Many of the examples are the observations of other mathematicians.

One example Guy gives is the conjecture that

2 ^{p-1} is prime—in fact, a Mersenne prime—when p is prime; but this conjecture, while true for p = 2, 3, 5 and 7, fails for p = 11 (and for many other values).

Another relates to the prime number race: primes congruent to 3 modulo 4 appear to be more numerous than those congruent to 1; however this is false, and first ceases being true at 26861.

A geometric example concerns Moser's circle problem (pictured), which appears to have the solution of

2 ^{n-1} for n points, but this pattern breaks at and above n =6 .

See also

• Insensitivity to sample size
• Law of large numbers (unrelated, but the origin of the name)
• Mathematical coincidence
• Pigeonhole principle
• Representativeness heuristic

External links

• Web site: Chris. Caldwell. Law of small numbers. The Prime Glossary.
  • Web site: Small finite sets. Secret Blogging Seminar. 2007-10-27. Scott. Carnahan., notes on a talk by Jean-Pierre Serre on properties of small finite sets..
• Belief in the law of small numbers. . . . Psychological Bulletin . 76 . 2 . 105–110. August 1971 . 10.1037/h0031322 . people have erroneous intuitions about the laws of chance. In particular, they regard a sample randomly drawn from a population as highly representative, I.e., similar to the population in all essential characteristics.. 10.1.1.592.3838 .

Notes and References

1. Guy . Richard K. . Richard K. Guy . 10.2307/2322249 . 8 . . 2322249 . 697–712 . The strong law of small numbers . 95 . 1988.
2. Gardner . Martin . Martin Gardner . December 1980 . Mathematical Games . 6 . . 24966473 . 18–28 . Patterns in primes are a clue to the strong law of small numbers . 243.