I'm working on a Mathematics paper at the moment, and my first finding was fairly specific. The next finding generalised the first finding, and my third finding generalised my second finding.

My first and second findings are therefore just special cases of my third finding, but they are still significant.

Is it appropriate for me to write all 3 findings in the paper and build upwards from the first, including the proofs for the the 1st and 2nd finding and generalising as we go, or should I just include the 3rd finding only, and point out afterwards that there are a couple of significant special cases that follow from it (namely finding 1 and finding 2)?

(Although including all 3 findings would lengthen the paper, I also feel that it would provide more of an insight into how I found result 3, rather than just mysteriously producing it from scratch.)

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    Ask your adviser. The details you’re giving are much too vague, it is unrealistic to expect that anyone could give you good advice based on just that. Basically the best way to present your findings could vary greatly based on the specific context, proof details, what’s already known etc, so there isn’t going to be a generic answer that fits all situations of this type. – Dan Romik Mar 9 at 20:57
  • @DanRomik I am still in high school, so I have no advisor :-( Sorry for being vague, I didn't realize, what further information would you like me to include? – A-Level Student Mar 9 at 20:58
  • I don’t think more detail would help, unless it’s a level of detail that isn’t suitable for this site. My advice is to find a professional mathematician you can show your work to, and ask for their advice. – Dan Romik Mar 9 at 21:06
  • @DanRomik isn't there a risk of plagiarism in that? – A-Level Student Mar 9 at 21:07
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    I can’t say. But even if so, working in isolation carries its own risks, which are likely greater. Talking to active researchers could save you many months of working on things that are trivial/already known/nominally new but not interesting/etc. – Dan Romik Mar 9 at 21:14

Either way could be made to work, but consider:

If the proof methods of the three are essentially the same, then the first way would be boring (possibly).

But if the second builds on the first (etc) by using different insights and methodologies, then the first way might well be superior.

Don't just consider the progression of results but of the progression of proofs and, especially, insights, that get you to the final result.

And if the latter is the case, make sure you emphasize those insights. They might (and often do) lead to developing other ideas.

  • Thank you for your advice; my paper fits into the first category, so the proofs are almost identical, just progressively more complex (by a small amount) each time. – A-Level Student Mar 9 at 21:00

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