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Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complete graph starts $0,1,3,6,10$, so the next term must be 15 etc."

Granted, this second statement is less logically unsound than the first since it's not difficult to see the reason why the sequence must continue as such; nevertheless, the statement was made on a premise that boils down to "interesting patterns must always continue".

I try to counter this logic by creating a ridiculous argument like "the numbers $1,2,3,4,5$ are less than $100$, so surely all numbers are", but this usually fails to be convincing.

So, are there any examples of non-trivial patterns that appear to be true for a large number of small cases, but then fail for some larger case? A good answer to this question should:

  1. be one which could be explained to the layman without having to subject them to a 24 lecture course of background material, and
  2. have as a minimal counterexample a case which cannot (feasibly) be checked without the use of a computer.

I believe conditions 1. and 2. make my question specific enough to have in some sense a "right" (or at least a "not wrong") answer; but I'd be happy to clarify if this is not the case. I suppose I'm expecting an answer to come from number theory, but can see that areas like graph theory, combinatorics more generally and set theory could potentially offer suitable answers.

  • 115
    The sentence: ""the numbers 1,2,3,4,5 are less than 100, so surely all numbers are" - Is interesting.2012-02-20
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    You might also be interested in this (MO thread) [http://mathoverflow.net/questions/11978/heuristically-false-conjectures] which brings up Merten's Conjecture and a few others.2012-02-20
  • 11
    @yasmar, I was thinking of this: http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples2012-02-20
  • 0
    Thanks @Gerry I had not seen that one. The one I linked to is not quite as relevant, but still related.2012-02-20
  • 26
    This doesn't satisfy b), but how about "$n^2-n+41$ is always prime"? (it's true for $1\le n\le 40$).2012-02-20
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    I once asked a similar-in-spirit question at MO: http://mathoverflow.net/questions/52101/longest-coinciding-pair-of-integer-sequences-known2012-02-21
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    Richard Guy wrote a couple of Monthly articles on this, calling it "The Strong Law of Small Numbers" in the 90s. [Here](http://www.math.sjsu.edu/~hsu/courses/126/Law-of-Small-Numbers.pdf) is one of them.2012-02-21
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    See also http://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples.2012-02-21
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    For a simple explanation take a rocket climbing up depending on the amount of fuel / thrust / gravity. It would go up slowly until gravity suddenly takes over when it runs out of fuel.2012-02-21
  • 3
    Since this is big list and there are no definitive answers, shouldn't it be community wiki?2012-02-21
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    @joriki: I would say this is a duplicate of that one, just with different wording.2012-02-21
  • 21
    @EmmadKareem After reading halfway through this page, this looks like a challenge to see who can give the most mind blowing example of this simplified version: "N not equals 82174583229565384923 for N = 1,2,3,4... breaks down at N = 82174583229565384923"2012-02-22
  • 0
    [Strong Law of Small Numbers](http://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers) is also mentioned at Wikipedia.2012-03-19
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    Dividing a sphere into two halves, does not reduces it's Total surface area to 50%. Does this statement makes any sense to you and appropriate for this question?2012-06-19
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    " I receive responses like: "surely if the Collatz Conjecture is true up to $20\times2^{58}$, then it must always be true?'; and "the sequence of number of edges on a complete graph starts $0,1,3,6,10$, so the next term must be $15$ etc"...hmmm, I don't think that you are talking to laymen after all.2013-02-17
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    Reading the answers to this question are very entertaining.2013-03-12
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    $e = 2.7 \, 1828 \, 1828 \quad $ :O $ \quad 459045235 \,$ :(2013-07-14
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    That's why I don't like problems that say 2,4,6,8,10,12,14,16 what is the next number? http://oeis.org/A0620282014-10-12

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