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. 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 as of 3 November 2012. Since November 2013, that number was 14228, at least until 14 April 2014. 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.
However, as there are many significant results in mathematics that make use of self-reference, the paradox illustrates some of the power of self-reference, 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".
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