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


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

Revision as of 06:02, 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.

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, does ∑ k = 1 n − 1 ( − 1 ) k − 1 ( n − k ) ! {\displaystyle \sum _{k=1}^{n-1}(-1)^{k-1}(n-k)!} always prime?
  • Define an: a0 = 1, for k > 0, ak = (1+a0+a1+...+ak−1)/k, the first few terms of an are 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, is an always integer?
  • Is 4 ⋅ 72 n − 1 {\displaystyle 4\cdot 72^{n}-1} composite for all n ≥ 1 {\displaystyle n\geq 1} ?
  • Is ∫ 0 ∞ ∏ k = 1 n cos ⁡ ( x k ) d x = π 2 {\displaystyle \int _{0}^{\infty }\prod _{k=1}^{n}\cos \left({\frac {x}{k}}\right)dx={\frac {\pi }{2}}} for all n ≥ 1 {\displaystyle n\geq 1} ?
  • Riemann's function L i ( x ) {\displaystyle Li(x)} is an approximation for π ( x ) {\displaystyle \pi (x)} (the number of primes less than or equal to x), is π ( x ) ≤ L i ( x ) {\displaystyle \pi (x)\leq Li(x)} always true?
  • Mertens function M ( n ) = ∑ 1 ≤ k ≤ n μ ( k ) {\displaystyle M(n)=\sum _{1\leq k\leq n}\mu (k)} , is | M ( n ) | < n {\displaystyle \left|M(n)\right|<{\sqrt {n}}} always true?

See also

Notes

  1. 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.
  2. Gardner, M. "Mathematical Games: Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243, 18-28, Dec. 1980.
  3. ^ Guy, Richard K. (1990). "The Second Strong Law of Small Numbers". Mathematics Magazine. 63 (1): 3–20. doi:10.2307/2691503. JSTOR 2691503.

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