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Let us call an integer n > 2 strong if the greatest prime divisor of n is greater than the greatest prime divisor of (n + 1). Then, obviously, every odd prime is a strong number, but there are also many composite strong numbers, the smallest of which is 14. (Whether the fact that 14 is the smallest strong number is related to the fact that the answer to Kuratowski’s closure-complement problem is 14 would itself be an interesting question.)

So, have what I am calling strong numbers already been given a name, and, in any case is there a pattern to them?

  • You had me until you brought in the Kuratowski thing. It's like saying it would be interesting to find out whether the moon is made of cheese because the moon is white and there is white cheese. – Matt Samuel Apr 23 '17 at 03:01
  • Verifying that $14$ is the smallest strong composite just involves checking the smaller composite numbers. There are not many of them. – Ross Millikan Apr 23 '17 at 04:18
  • @RossMillikan: Thanks, but you’re picking up the wrong end of the stick. The relationship would perhaps go something like this: ‘Because of the nature of the closure-complement scenario, the maximum number must be composite, moreover strong, moreover minimal, and hence 14.’ A similar situation is perhaps the open problem as to the smallest algebra in which Tarski’s 11 high school axioms hold, but for which the identity W(x,y) fails. The cardinality, according to the Wikipedia article on ‘Tarki’s high school algebra problem’ is known to be 11 or 12. –  Apr 25 '17 at 20:07
  • The word "strong number" has another meaning: a number whose digits' factorials sum to itself. E.g., $145$ is a "strong number" under this definition, because $$1! + 4! + 5! = 1 + 24 + 120 = 145.$$ – L. F. May 02 '19 at 08:38

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