Interesting number paradox

The interesting number paradox is a semi-humorous paradox which arises from the attempt to classify every natural number as either "interesting" or "uninteresting". The paradox states that every natural number is interesting. The "proof" is by contradiction: if there exists a non-empty set of uninteresting natural numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, thus producing a contradiction.

In a discussion between the mathematicians G. H. Hardy and Srinivasa Ramanujan about interesting and uninteresting numbers, Hardy remarked that the number 1729 of the taxicab he had ridden seemed "rather a dull one", and Ramanujan immediately answered that it is interesting, being the smallest number that is the sum of two cubes in two different ways.

Paradoxical nature

Attempting to classify all numbers this way leads to a paradox or an antinomy of definition. Any hypothetical partition of natural numbers into interesting and uninteresting sets seems to fail. Since the definition of interesting is usually a subjective, intuitive notion of "interesting", it should be understood as a half-humorous application of self-reference in order to obtain a paradox.

The paradox is alleviated if "interesting" is instead defined objectively: for example, the smallest integer that does not appear in an entry of the On-Line Encyclopedia of Integer Sequences was originally found to be 11630 on 12 June 2009.[1] The number fitting this definition later became 12407 from November 2009 until at least November 2011, then 13794 as of April 2012, until it appeared in sequence OEIS: A218631 as of 3 November 2012. Since November 2013, that number was 14228, at least until 14 April 2014.[1] (This definition of uninteresting is possible only because the OEIS lists only a finite number of terms for each entry. For instance, OEIS: A000027 is the sequence of all natural numbers, and if continued indefinitely would contain all positive integers. As it is, the sequence is recorded in its entry only as far as 77.) Depending on the sources used for the list of interesting numbers, a variety of other numbers can be characterized as uninteresting in the same way.[2]

However, as there are many significant results in mathematics that make use of self-reference (such as Gödel's incompleteness theorems), the paradox illustrates some of the power of self-reference[3], and thus touches on serious issues in many fields of study.

The mathematician and philosopher Alex Bellos suggested in 2014 that a candidate for the lowest uninteresting number would be 247 because it was, at the time, "the lowest number not to have its own page on Wikipedia".[4]

Website smallest number not in this website
Wikipedia 262 (or 275, as 275 is the smallest number having no properties in Wikipedia except prime factorization)
What's Special About This Number? 391
Prime curios 492 (326 for approved curios)
Properties of first 5000 integers 291
Number properties 309
Number properties 80
OEIS 17843

See also

Further reading
  • Gardner, Martin (1959). Mathematical Puzzles and Diversions. ISBN 0-226-28253-8.
  • Gleick, James (2010). The Information (chapter 12). New York: Pantheon Books. ISBN 978-0-307-37957-3.

References
  1. Johnston, N. (June 12, 2009). "11630 is the First Uninteresting Number". Retrieved November 12, 2011.
  2. Charles R Greathouse IV. "Uninteresting Numbers". Retrieved 2011-08-28.
  3. (See, for example, Gödel, Escher, Bach#Themes, which itself -- like this section of this article -- [also] mentions, -- and contains a wikilink to! -- [the article about] "self-reference".)
  4. Bellos, Alex (June 2014). The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life. illus. The Surreal McCoy (1st Simon & Schuster hardcover ed.). N.Y.: Simon & Schuster. pp. 238 & 319 (quoting p. 319). ISBN 978-1-4516-4009-0.
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