I got a rejection recently on a short paper that gave a fairly short proof that a conjecture of a well-known mathematician was false. The reviewer said that “while the results are new and interesting,” “the proofs are fine, but not very difficult, and the techniques are not new.” Nonetheless, the arguments previously evaded some top people in the field who were interested in the topic. My question is, to what kinds of mathematics journals should one submit significant results whose proofs involve some clever trick rather than a revolutionary method? Also, should one not strive to find the most transparent form of a mathematical argument?
There's a good chance the rejection is because your results are not interesting enough for the journal. In other words, the journal is looking for not only new and interesting results, but also complicated proofs and/or new techniques.
So: submit to another, probably less prestigious journal. Alternatively, you can submit to a similarly-ranked journal and hope the peer review process results in acceptance, which is entirely possible, because this particular reject reason is a judgement call and different people will come to different recommendations.
Keep submitting to other journals. If the result is indeed significant then you will eventually find a journal that will want to publish it. If on the other hand you continue to consistently get more rejections, that probably means the result is not as significant as you seem to think. Keep in mind that just because a well known mathematician conjectures something, that does not automatically make it interesting, and does not automatically mean that your counterexample “evaded top people in the field”.
I would suggest that you focus your thoughts less on "the proofs are [...] not very difficult" and more of your thoughts on "the techniques are not new." What happened here is that the referee doesn't think your paper is interesting enough for the journal that you submitted to because it doesn't contain important new ideas. This is a perfectly valid reason to reject a paper from a strong journal.
The first thing to think about is whether you agree with the referee here, do you think that the paper does involve substantial new ideas? If you think there are new ideas, then rewrite the introduction so that it will be clear to a referee that there are new ideas and then resubmit to another journal of the same caliber. If you agree with the referee and think there aren't new ideas, then leave the paper alone and resubmit to a less competitive journal. The referee clearly thinks that your paper would be publishable somewhere, so if you just lower your aims a little you shouldn't have trouble publishing it.
Finally two random bits of advice. First, if the paper is quite short you might consider a journal that focuses on short papers (like Proceedings of the AMS). Second, make sure that your intro explicitly says "this is an open problem posed by X in year Y." If it's only been open for a year or two and it's solvable by a standard method, then that means it just wasn't as interesting a problem as the person who posed it thought it was. But if it's been open for 10 years, then that's evidence your application of these techniques is more clever than it looks at first glance.
You could perhaps emphasize the importance of your proof, therefore making your paper more about the result than the proof itself.
Sometimes, some writing judo (turning the disadvantages into advantages) can also help. Instead of "the proof is simple" and "the techniques are not new", maybe the proof is "elegant in its simplicity" and uses "well-established techniques". Being up-front with it allows you to control the narrative. Of course, this type of reframing can only take you so far...
This sounds to me like the reviewer provided a shoddy argumentation for their judgment. How very dare you make your proof easy to read, instead of obfuscating it into a complex unreadable mess that makes it look more impressive?
If I were you, I'd resubmit to a similar-caliber journal, and hope for a more fortuitous pull of the one-armed reviewer bandit. Alternatively, you could consider corresponding about this issue with the editor of the journal from which you got this review. Surely, they would agree that making proofs difficult shouldn't be a goal in itself, so if that were the only argument for rejection you'd have a good case for being reconsidered.
You could talk to the famous mathematician who made the conjecture. He could be more resourceful than you and he might know the proper journal for your contribution. Moreover, he seems to be the only one who can endorse your contribution.
Edit: if he is an editor, you can also submit the paper to him if the paper align with the journal's coverage.
He will be more than happy to read your paper and provide you feedbacks because your proof is smart and concise.
To complement the other answers: on one hand, yes, I have seen several instances where a proof (of a significant result) turned out to be "insufficiently impressive/interesting", and was rejected. On another hand, yes, this can mean that an interesting meta-fact was discovered, namely, that the result turns_out to be (construable as!!!) so easy that it's abruptly not-so-interesting. Yes, a bit ironic.
On a third hand, there is a possibility that some state-of-the-art methods, possibly not widely known, render a question "routine", even though to the perception of many it is not obviously answerable at all. To greatly exaggerate: not all products of large integers are "known/documented", but those of us hip to the grade-school algorithm (or better) will not be interested at all in seeing any particular product explained (without some further features of interest). So the "new" feature is there, but it's insufficient...
Unsurprisingly, in the end, there is considerable context-dependence/subjectivity in referees' appraisals of submissions. And, indeed, to some degree, editors care about whether a given paper will enhance, versus diminish, the "reputation" of the journal. "Impressive" papers are sometimes preferred... And, again, this is context-dependent and subjective.
So, yes, sometimes a (meta?) proof that an issue is, after all, (potentially) straightforward often does not lend itself to "publishability". :)
EDIT: I added the quotes to "publishability" because, after all, putting a document on the internet is a far more powerful form ofliteral publishing than anything achieved prior to 1985 or so. Yes, I know we're talking about various forms of status-certification and gate-keeping. Yes, some quality control also, but not only...
What you are discovering is that, in many cases, the proofs are more important than the theorems, especially when they bring insight or when there is a new technique in the proof that might be applied to other work.
It seems like your work is "a bit interesting" but doesn't have the potential impact to bother with publishing it. Especially when seen in competition with other papers.
For my doctoral research I worked on three problems. For one it was so easy to postulate and prove theorems (several per week) that it was abandoned as trivial. It was "cute" but, as they say, there was no "there" there.
You likely gained some insight in the work, but if the proofs were trivial, others before you likely did also, but didn't think it was worth publishing.
The comment of Joel Reyes Noche to transform it into a short note is a good one, as are the comments and answers of others to just try somewhere else are also possibilities.
But another possibility is to take what insight you got from this work and use it to extend to something with more potential impact.
The now current XKCD might apply here.