In
mathematics, the "strong law of small numbers" is the humorous law that proclaims, in the words of
Richard K. Guy (1988):
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
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...
.
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
Lester is an ancient Anglo-Saxon surname and given name. Notable people and characters with the name include:
People
Given name
* Lester Bangs (1948–1982), American music critic
* Lester W. Bentley (1908–1972), American artist from Wisc ...
.)
Second strong law of small numbers
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 is prime—in fact, a ]Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
—when is prime; but this conjecture, while true for = 2, 3, 5 and 7, fails for = 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 for points, but this pattern breaks at and above .
See also
* Insensitivity to sample size
Insensitivity to sample size is a cognitive bias that occurs when people judge the probability of obtaining a sample statistic without respect to the sample size. For example, in one study subjects assigned the same probability to the likelihood of ...
* Law of large numbers (unrelated, but the origin of the name)
* Mathematical coincidence
A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.
For example, there is a near-equality close to the round number 1000 between powers ...
* Pigeonhole principle
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there mu ...
* Representativeness heuristic The representativeness heuristic is used when making judgments about the probability of an event under uncertainty. It is one of a group of heuristics (simple rules governing judgment or decision-making) proposed by psychologists Amos Tversky and Da ...
Notes
External links
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Mathematics papers
Mathematical humor
1988 documents
1988 in science
Works originally published in American magazines
Works originally published in science and technology magazines
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