View source for Strong law of small numbers

{{short description|Humorous mathematical law}}
{{Other uses|Law of small numbers (disambiguation)}}

In [[mathematics]], the "'''strong law of small numbers'''" is the [[humor]]ous law that proclaims, in the words of [[Richard K. Guy]] (1988):<ref>{{cite journal
 | last = Guy | first = Richard K. | author-link = Richard K. Guy
 | doi = 10.2307/2322249
 | issue = 8
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2322249
 | pages = 697–712
 | title = The strong law of small numbers
 | url = https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Guy697-712.pdf
 | volume = 95
 | year = 1988}}</ref>
{{quote|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. Earlier (1980) this "law" was reported by [[Martin Gardner]].<ref>{{cite journal
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | date = December 1980
 | department = Mathematical Games
 | issue = 6
 | journal = [[Scientific American]]
 | jstor = 24966473
 | pages = 18–28
 | title = Patterns in primes are a clue to the strong law of small numbers
 | volume = 243}}</ref> 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==
[[file:circle_division_by_chords.svg|thumb|Guy gives [[Moser's circle problem]] as an example. The number of {{nowrap|points (''n''),}} {{nowrap|chords (''c'')}} and {{nowrap|regions (''r<sub>G</sub>'')}}. 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''':
{{quote|When two numbers look equal, it ain't necessarily so!<ref name=SSL>{{cite journal
 | last = Guy | first = Richard K. | author-link = Richard K. Guy
 | doi = 10.2307/2691503
 | issue = 1
 | journal = [[Mathematics Magazine]]
 | jstor = 2691503
 | pages = 3–20
 | title = The second strong law of small numbers
 | volume = 63
 | year = 1990}}</ref>}}

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.<ref name=SSL/>

One example Guy gives is the conjecture that <math>2^p-1</math> is prime—in fact, a [[Mersenne prime]]—when <math>p</math> is prime; but this conjecture, while true for <math>p</math> = 2, 3, 5 and 7, fails for  <math>p</math> = 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 <math>2^{n-1}</math> for <math>n</math> points, but this pattern breaks at and above <math>n=6</math>.

==See also==
* [[Insensitivity to sample size]]
* [[Law of large numbers]] (unrelated, but the origin of the name)
* [[Mathematical coincidence]]
* [[Pigeonhole principle]]
* [[Representativeness heuristic]]

==Notes==
{{Reflist}}

==External links==
* {{Cite web
|first=Chris
|last=Caldwell
|title=Law of small numbers
|url=http://primes.utm.edu/glossary/page.php?sort=LawOfSmall
|work=The Prime Glossary
}}
* {{MathWorld|urlname=StrongLawofSmallNumbers|title=Strong Law of Small Numbers}}
* {{Cite web
|title=Small finite sets
|work=Secret Blogging Seminar
|date=2007-10-27
|first=Scott
|last=Carnahan
|url=http://sbseminar.wordpress.com/2007/10/27/small-finite-sets/
|postscript=, notes on a talk by [[Jean-Pierre Serre]] on properties of small finite sets.
}}
* {{cite journal |title=Belief in the law of small numbers. |author1=[[Amos Tversky]] |author2=[[Daniel Kahneman]] |journal=Psychological Bulletin |volume=76 |number=2 |pages=105–110|date=August 1971 |doi=10.1037/h0031322 |quote=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.|citeseerx=10.1.1.592.3838 }}

[[Category:Mathematics papers]]
[[Category:Mathematical humor]]
[[Category:1988 documents]]
[[Category:1988 in science]]
[[Category:Works originally published in American magazines]]
[[Category:Works originally published in science and technology magazines]]


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