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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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    @MarianoSuárez-Alvarez: It was my way of 1) rewarding the early bird and, more importantly 2) of closing the question, realizing that Adam Smith was right when he said that this question is "far too broad to generate useful answers". If you want to convert this to CW and continue with a big list, that would be ok by me:)2011-10-22

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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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    Uh, I was hoping for some explicit examples:)2011-10-21