Strong law of small numbers
In mathematics, the "strong law of small numbers" is the humorous law that proclaims, in the words of Richard K. Guy (1980, 1988):[1][2]
There aren't enough small numbers to meet the many demands made of them.
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. Guy's 1988 paper gives numerous examples in support of this thesis and earned him the AMA Lester R. Ford Award in 1989.[3]
Second strong law of small numbers
[edit]
Guy also formulated a second strong law of small numbers:
When two numbers look equal, it ain't necessarily so![4]
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.[4]
One example Guy gives is the conjecture that 2p − 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, 23, 29, ....
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 2n−1 for n points, but this pattern breaks at and above n = 6.
See also
[edit]- Insensitivity to sample size
- Law of large numbers (unrelated, but the origin of the name)
- Mathematical coincidence
- Pigeonhole principle
- Representativeness heuristic
Notes
[edit]- ↑ Guy, Richard K. (1988). "The strong law of small numbers" (PDF). The American Mathematical Monthly. 95 (8): 697–712. doi:10.2307/2322249. JSTOR 2322249. Archived from the original (PDF) on 13 August 2021.
- ↑ Unpublished paper by Richard Guy quoted in Gardner, Martin (December 1980). "Patterns in primes are a clue to the strong law of small numbers". Mathematical Games. Scientific American. 243 (6): 18–28. doi:10.1038/scientificamerican1280-18. JSTOR 24966473.
- ↑ Happy Birthday, Richard Guy! Archived 25 April 2017 at the Wayback Machine Mathematical Association of America, 30 September 2016
- 1 2 Guy, Richard K. (1990). "The second strong law of small numbers". Mathematics Magazine. 63 (1): 3–20. doi:10.2307/2691503. JSTOR 2691503.
External links
[edit]- Caldwell, Chris. "Law of small numbers". The Prime Glossary.
- Weisstein, Eric W. "Strong Law of Small Numbers". MathWorld.
- Carnahan, Scott (2007-10-27). "Small finite sets". Secret Blogging Seminar, notes on a talk by Jean-Pierre Serre on properties of small finite sets.
{{cite web}}: CS1 maint: postscript (link) - Amos Tversky; Daniel Kahneman (August 1971). "Belief in the law of small numbers". Psychological Bulletin. 76 (2): 105–110. CiteSeerX 10.1.1.592.3838. doi: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.
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