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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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I'm not sure I understand the question. Aren't many open questions themselves of interest? –  Qiaochu Yuan Oct 21 '11 at 21:35
    
Let me rephrase: are there any known connections between famous open questions, and perhaps also some famous theorems? A historical example would be the Taniyama–Shimura–Weil conjecture (now theorem) and Fermat's last theorem. –  Yuval Filmus Oct 21 '11 at 21:41
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The Clay Mathematics Institute has prizes for a bunch of unsolved problems that all have further implications. claymath.org/millennium –  Thomas Andrews Oct 21 '11 at 23:09
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@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! –  Mariano Suárez-Alvarez Oct 22 '11 at 0:05
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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:) –  Mike Jones Oct 22 '11 at 3:20

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up vote 3 down vote accepted

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:) –  Mike Jones Oct 21 '11 at 22:40

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