How to propose a conjecture

What are the basic things (about when and how) to be kept in mind while proposing a conjecture in Mathematics. Should it accompany solid efforts at proof. When should any one think of proposing a conjecture.I'd like to know some of these details just out of curiosity.

First of all, there are various attitudes towards conjectures. Let me describe two rather famous instances before making some remarks:

I'd like to start with "Serre's conjecture".

In his famous paper Faisceaux Algébriques Cohérents (often called simply FAC) the Fields medalist J.-P. Serre mentioned a specific technical problem:

Loose translation:

Let us point out that, if $V = K^r$ (in which case $A = K[X_1,\cdots,X_r]$), we don't know whether there exist finitely generated projective $A$-modules that are not free, or, what amounts to the same, whether there exist non-trivial algebraic vector bundles with base $K^r$.

Here is the very first paragraph of T. Y. Lam's book entitled Serre's conjecture (taken from Springer LNM 635):

On P.243 of his famous article Faisceaux algébriques cohérents (FAC), Serre wrote: 'On ignore s'il existe des $A$-modules projectifs de type fini qui ne soient pas libres' ( $A = k[t_1,\ldots,t_n]$, $k$ a field); shortly thereafter, the freeness of finitely generated projective modules over $k[t_1,\ldots,t_n]$ became known to the mathematical world as "Serre's Conjecture". Serre might have objected to the fact that what he raised essentially as an open problem became his "conjecture" by world acclamation, but the fine distinction between "Serre's Problem" and "Serre's Conjecture" may now be safely left to the deliberation of the mathematical historian. Culminating almost twenty years of effort by algebraists, D. Quillen and A. Suslin proved independently, in January, 1976, that finitely generated projective modules over $k[t_1,\ldots,t_n]$ are, indeed, free.

The second instance I'd like to mention appears in Wolfgang Lück's work frequently. He is an internationally acclaimed topologist whose recent work centers around the Baum-Connes Conjecture and related questions. A quick search on the page containing his list of publications reveals that the word "conjecture" appears no less than 42 times. To be fair, I should note that a good many of them are titles of talks and conference names, but nevertheless the word conjecture is used quite often in his surroundings. The talks I saw by him always mentioned a few of these conjectures and also proved a few instances.

Here's a random sample from his Inventiones paper The relation between the Baum-Connes Conjecture and the Trace Conjecture

and a bit further down:

This should illustrate two rather diametrically opposed attitudes to "conjectures" by two very famous mathematicians. Serre rather relucantly pointed out an open problem which was considered so interesting that it became his conjecture, more to his dismay than to his joy, I might add. Its resolution by Quillen certainly contributed to his earning the Fields Medal while Suslin didn't receive it for reasons somewhat unclear to me, but that's a different story...

On the other hand, Lück uses the word conjecture on a daily basis and the conjectures he talks about and his results concerning them are considered very interesting and ground-breaking by many people (including me of course).

Bear in mind however that both Lück and Serre are mathematicians known world-wide and more than deservedly so.

For more modestly talented people like you and me I think it is far more appropriate to follow Serre's example and content ourselves just hinting at open problems. It is probably a safer bet to assume that neither you nor me will ever phrase a question considered important, interesting and difficult enough by many enough to get elevated to the status of conjectures. Running ahead and throwing around conjectures is simply not an appropriate attitude I think. But that's just a piece of unsolicited advice and my two cents.

So far, I haven't really addressed your question "how to propose a conjecture". My response is simply: "Don't."

(But see Matt E's answer for reasons why formulating conjectures may be justified, appropriate and useful).

Here's a brief explanation of my advice:

In order for a conjecture to be taken seriously a good many sociological conditions must be fulfilled:

• You must be taken seriously by your audience.
• You must know what you're talking about.
• You must be convinced that the conjecture is an important problem (not only you must consider it important but others should be likely to agree).
• You must be sure that the conjecture is unlikely to produce silly counter-examples or be confirmed quickly and easily.

If you formulate a conjecture, this might be interpreted to say that you're thinking that all these conditions are fulfilled. Out of reasons of modesty I think it is therefore best to refrain from making them, at least as long as you haven't reached a certain status of recognition in your research area.

After having made some contributions to your field you will probably be in position to judge whether formulating a conjecture rather than a question or a problem is appropriate or not, for instance for reasons as listed by Matt E in his answer. I am not in position to give any advice on that.

Added: joriki made a good point that deserves to be mentioned more prominently:

[...] I wholly agree with your thoughts on modesty. However, it seems you're treating the word "conjecture" as synonymous with "(potentially) famous conjecture", "conjecture (potentially) named after someone". I think it's not immodest to use it in another sense, as in "There is an obvious pattern here that suggests the conjecture that ... for all n." That doesn't have to imply that you think this is a problem that will raise the interest of generations of mathematicians and advance mathematics immeasurably.

For instance, Popper uses the word in this more mundane sense in his "Conjectures and Refutations", e.g.: "The actual procedure of science is to operate with conjectures: to jump to conclusions -- often after a single observation [...]."

I do see these points and objections. I don't have anything much to add or object to that since I don't know Popper's writings very well. Thank you for raising this point joriki.

Rajesh, you are of course free to follow this interpretation but take my answer as a word of warning that some people tend to think of conjectures the way I tried to express it above. The word "conjecture" is loaded and it is better to be safe than sorry.

Let me display an excerpt from the text Some Hints on Mathematical Style by David Goss:

I'll finish by recommending reading these hints (based on remarks by Serre) in full. Read them attentively and think about the points raised there. You may or may not agree with all the points but I think it should be profitable in any case.

I think you should prove a few theorems before you propose any conjectures. Better yet, publish a few papers. Anyone can make conjectures; yours won't stand out from the others until people think something coming from you is worth looking at.