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The thing of interest need not be an unproven result. For (an off-the-wall) example, does the Collatz Conjecture imply Fermat’s Last Theorem?

Of course, by “imply” here, we really mean “entail”. If something is known to be true, then ANYTHING implies it in a truth-table sense, but that is not what is meant, of course.

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    $\begingroup$ I'm not sure I understand the question. Aren't many open questions themselves of interest? $\endgroup$ Oct 21, 2011 at 21:35
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    $\begingroup$ @Yuval Filmus: I sort of shot myself in the foot with my naive wording, as Qiaochu Yuan pointed out, and so I appreciate your rephrasing very much. $\endgroup$
    – Mike Jones
    Oct 21, 2011 at 22:36
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    $\begingroup$ The Clay Mathematics Institute has prizes for a bunch of unsolved problems that all have further implications. claymath.org/millennium $\endgroup$ Oct 21, 2011 at 23:09
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    $\begingroup$ I suspect that this question is far too broad to generate useful answers. The set of open questions and the chains of implications between them are far too vast... $\endgroup$
    – Adam Smith
    Oct 21, 2011 at 23:49
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    $\begingroup$ @Mike, if you where looking for examples, it is a bit weird to accept one answer —the first one providing actual examples!— two hours after asking the question. We will never know what examples people who are sleeping right now due to the shape of the earth have available! $\endgroup$ Oct 22, 2011 at 0:05

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The abc conjecture has quite a number of interesting implications.

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Yes, probably all of them do. Attempts have probably been made to disprove these open questions by showing that they imply interesting things and then proving that these interesting things are not true. The first of these two steps has probably been taken in some form for every interesting open problem.

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  • $\begingroup$ Uh, I was hoping for some explicit examples:) $\endgroup$
    – Mike Jones
    Oct 21, 2011 at 22:40

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