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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.

Mike Jones
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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
  • The Collatz conjecture may not itself be of more than recreational interest (but what do I know?) -- but I'd definitely say that something such as $P=NP?$ is inherently interesting. – hmakholm left over Monica Oct 21 '11 at 21:41
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    @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. – Mike Jones Oct 21 '11 at 22:36
  • @Andre Nicolas: You don't seem to have read the question. I explicity excluded the requirement that assertion B be unproven. However, your phrase "quite different looking" is a welcome edit. By the same token, however, the example I gave of the Collatz Conjecture and FLT was a good one:) – Mike Jones Oct 21 '11 at 22:39
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    The Clay Mathematics Institute has prizes for a bunch of unsolved problems that all have further implications. http://www.claymath.org/millennium/ – Thomas Andrews Oct 21 '11 at 23:09
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    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... – Adam Smith Oct 21 '11 at 23:49
  • Since Fermat's last theorem is a theorem, so is "P implies FLT" for any statement P. – Chris Eagle Oct 21 '11 at 23:49
  • @Chris Eagle: but not "P ENTAILS FLT". Did you not read the entire question? – Mike Jones Oct 21 '11 at 23:54
  • @Mike: So what does "entail" mean? – Chris Eagle Oct 21 '11 at 23:57
  • @Chris Eagle: "Entail" means "obtain by means of elbow-grease, without the use of mirrors" :) – Mike Jones Oct 22 '11 at 00:02
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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-Álvarez Oct 22 '11 at 00:05
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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 03:20

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The abc conjecture has quite a number of interesting implications.

Fernando Martin
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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.

opt
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