Strong law of small numbers: Difference between revisions

Browse history interactively ← Previous edit Next edit →Content deleted Content addedVisual WikitextInline

Revision as of 08:37, 2 March 2021 edit御犽真夜魂 (talk \| contribs)187 editsNo edit summaryTag: Reverted← Previous edit		Revision as of 08:39, 2 March 2021 edit undoSalix alba (talk \| contribs)Edit filter managers, Administrators26,099 editsm Reverted edits by 御犽真夜魂 (talk) to last version by Salix albaTag: Rollback Next edit →
Line 46:		Line 46:
	Guy explains the latter law by the way of examples: he cites numerous sequences for which observing a subset of 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/>		Guy explains the latter law by the way of examples: he cites numerous sequences for which observing a subset of 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/>

			Some examples Guy gives include the ], ] where <math>2^p-1</math> is prime when <math>p</math> is one of the first few primes but fails for <math>2^{11}-1</math>, and the ] where primes congruent to 3 modulo 4 appear to be more numerous that those congruent to 1, this first fails for primes less than 26861.
	== Examples ==

	* 3!−2!+1! = 5, 4!−3!+2!−1! = 19, 5!−4!+3!−2!+1! = 101, 6!−5!+4!−3!+2!−1! = 619, 7!−6!+5!−4!+3!−2!+1! = 4421, they are all primes, is <math>\sum_{k=1}^{n-1} (-1)^{k-1}(n-k)!</math> always prime?

	* Define ''a''<sub>''n''</sub>: ''a''<sub>0</sub> = 1, for ''k'' > 0, ''a''<sub>''k''</sub> = (1+''a''<sub>0</sub><sup>2</sup>+''a''<sub>1</sub><sup>2</sup>+...+''a''<sub>''k''−1</sub><sup>2</sup>)/''k'', the first few terms of ''a''<sub>''n''</sub> are 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, is ''a''<sub>''n''</sub> always integer?

	* Is <math>4 \cdot 72^n-1</math> composite for all <math>n \ge 1</math>?

	* Is <math>\int_0^\infty \prod_{k=1}^n \cos \left( \frac{x}{k} \right) dx = \frac{\pi}{2}</math> for all <math>n \ge 1</math>?

	* Riemann's function <math>Li(x)</math> is an approximation for <math>\pi(x)</math> (the number of primes less than or equal to ''x''), is <math>\pi(x) \le Li(x)</math> always true?

	* Look at the remainders when the first few primes are divided by four: 2, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 1, 1, 3, 3, 1, 1, ..., it looks like if we stop this list at any point, there are always at least as many 3's as there are 1's, but is this always true?

	* Mertens function <math>M(n) = \sum_{1\le k \le n} \mu(k)</math>, is <math>\left\| M(n) \right\| < \sqrt { n }</math> always true?

	==See also==		==See also==
	* ]		* ]
Line 80:		Line 65:
	\|work=The Prime Glossary		\|work=The Prime Glossary
	}}		}}
	* , R. K. Guy, Amer. Math. Monthly 95 (1988), no. 8, 697-712.
	* {{MathWorld\|urlname=StrongLawofSmallNumbers\|title=Strong Law of Small Numbers}}		* {{MathWorld\|urlname=StrongLawofSmallNumbers\|title=Strong Law of Small Numbers}}
	* {{Cite web		* {{Cite web

Revision as of 08:39, 2 March 2021

Humorous mathematical law For other uses, see Law of small numbers (disambiguation).

In mathematics, the "Strong Law of Small Numbers" is the humorous law that proclaims, in the words of Richard K. Guy (1988):

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. Guy's paper gives numerous examples in support of this thesis.

Guy also formulated the Second Strong Law of Small Numbers:

When two numbers look equal, it ain't necessarily so!

Guy explains the latter law by the way of examples: he cites numerous sequences for which observing a subset of 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.

Some examples Guy gives include the Moser's circle problem, Mersenne prime where $2^{p}-1$ is prime when $p$ is one of the first few primes but fails for $2^{11}-1$ , and the Prime number race where primes congruent to 3 modulo 4 appear to be more numerous that those congruent to 1, this first fails for primes less than 26861.

Notes

Guy, Richard K. (1988). "The Strong Law of Small Numbers" (PDF). American Mathematical Monthly. 95 (8): 697–712. doi:10.2307/2322249. ISSN 0002-9890. JSTOR 2322249. Retrieved 2009-08-30.
Gardner, M. "Mathematical Games: Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243, 18-28, Dec. 1980.
^ 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

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.

This article about a mathematical publication is a stub. You can help Misplaced Pages by expanding it.

Categories:

Revision as of 08:39, 2 March 2021

See also

Notes

External links