# What are some interesting results that could be derived if some conjectures were true/false?

I recently came up on Djikstra's idea of a fictional company called Mathematics.Inc which produced proofs as trade secrets which could then be used by customers.. For example, it produced the proof of Reimann's Hypothesis and all mathematicians that used the Reimann Hypothesis for deriving interesting results had to pay royalty to the comany. The proof, however, was a trade secret.

So I was wondering if there have been mathematicians that have worked on finding interesting results that depend on unverified conjectures.. What are some interesting results that have been derived like this?

Some clarifications :

1. It isn't necessary that the conjecture is still not proven today. What matters is that it wasn't proved when those results were derived.

2. The results don't necissarily have to be true. For example, the results may have been derived assuming a conjecture is true and it may have later been found out that the conjecture is false. Sometimes, a ridiculous corollary may, in itself, have been the dis-proof of the conjecture.

3. This question is aimed, specifically, at results that were derived assuming the answer to an unresolved question. The mathematicians must know that the question is unresolved. It shouldn't be the case that they assumed something that was taken for granted then but is not rigorous today. I'm not looking for proofs in which potholes with assumptions were found later. For example, results proven assuming Hodge's conjecture, P vs NP or Fermat's Last Theorem before it was proven, are the kind of answers I am looking for. But, a mathematician who found the sum of an interesting infinite series without proving it's convergence because the concept of convergence hadn't been defined yet is not the kind of answer I'm seeking. In the former case, the mathematician explicitly assumes something he knows is moot. But, in the latter... It's just the knowledge of the day that's inhibiting him. It's something taken for granted, not exactly an assumption.

• Pretty much all of cryptography relies on the existence of one-way functions, which is a conjecture (and for instance requires $\mathsf{P}\neq \mathsf{NP}$). More generally, 99% of complexity theory lower bounds rely on such assumptions. – Clement C. Mar 12 '16 at 2:53
• You may want to read this for instance. – Clement C. Mar 12 '16 at 2:57
• The idea of showing that you can prove something without giving the proof isn't as hypothetical as you might think, just look up zero information proofs. – DanielV Mar 12 '16 at 2:58
• @DanielV You mean Zero-Knowledge Proofs? – Clement C. Mar 12 '16 at 2:59
• @ClementC. Ah yes that is probably the more common term. – DanielV Mar 12 '16 at 3:17