Suppose a situation when two students trying to apply for a PhD position in pure mathematics and they both have same GPA, same Rec.Letters, etc except for their published papers. Let both applicants have one paper published each, but applicant A has a "much better" research done compared to applicant B.

What makes applicant A's paper "much better"?

If it is the number of people cited his/her paper in their references it is absurd because time is the main factor of growth of its credit.

If it is the reputation of journal the paper is published it is also not a good judgement because the very same journal can contain both proof to FLT and a PhD paper.

I don't think the length of paper matters... I don't think the time that a researcher has spent matters since no one asks that...

This is a question of comparing two research paper which neither contains a solution to a famous unsolved problem; it's about papers that normally publish by students (or even professors).

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    You probably need to specify an area or discipline. In my area, if a student has a good research paper, I usually assumes he/she has had help from his/her supervisor. Also, I factor in the tool, whether it has been applied in a sophisticated manner, and difficulty of the problem. For the tool, I determine whether such a tool will help with the problems in my area. If so, I will pick that student over another without the tool. – Prof. Santa Claus Sep 23 '18 at 4:14
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    I'd recommend reading Pete L. Clark's answer to the related question "What makes a research paper low-quality?". – Anyon Sep 23 '18 at 15:36
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    same GPA, same Rec.Letters, etc except for their published papers — If two applicants have different published papers but identical recommendation letters, then something is very wrong. Recommendation letters are supposed to be personalized to the applicant. In particular, given that these hypothetical applicants have publications, useful recommendation letters will not only describe their papers in some technical detail, but will actually offer value judgements about them. – JeffE Sep 24 '18 at 17:39

The quality of a mathematical paper isn't an absolute. It may not even be apparent for years. Quality is judged by individuals and two judges may come to opposite conclusions on the value.

Number of citations is no more than an indication. A person working in an especially arcane sub-field with few other researchers will naturally have fewer citations than someone working in a popular field. Who cites your work may mean more, actually.

The quality of the journal in which you find something is an indication, but partly that is a matter of coincidence. Perhaps the paper in the "lesser" journal is, again, in too arcane a field for the editors to have included it. It isn't that this means nothing, it is just that it isn't an absolute indication.

If the purpose of the question is anything beyond a hypothetical, related to who is more likely to advance, then that really depends on other factors. Certainly one paper isn't a great indicator. As a comment by Prof. Santa Claus notes, a future advisor is more likely to want to work with the student who has more to contribute to the actual work of the subfield of that advisor. Something in either paper may help that person to make a decision, but other factors will also come in to play.

This doesn't, of course, address the issue that some problems in mathematics are considered important. You mention famous unsolved questions of course. Giving a first correct proof of a problem already considered important is, of course, quality work.

  • Offtopic, but would you tell me if I can publish a paper in a journal which is 'based' in another country? For example my university is in Montenegro, can I publish in Australasian Journal of Combinatorics? Thank you. – Emma Sep 23 '18 at 15:35
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    @Emma, yes, of course. There are very few national constraints as long as the work doesn't involve national security of one country. For combinatorics, there would be no issue at all. – Buffy Sep 23 '18 at 15:39
  • a future advisor is more likely to want to work with the student who has more to contribute to the actual work of that advisor — This sounds a little off to me. Potential advisors are more likely to work with student whose interests match theirs, but not because it helps their research, but rather because those students are more likely to succeed. (I take it as written that most faculty cannot choose not to work with students at all.) – JeffE Sep 24 '18 at 17:43
  • @JeffE, I made a clarifying edit. You are right in most cases, but some students join labs in which they contribute to ongoing projects. – Buffy Sep 24 '18 at 19:56

This may be a bit of an odd reply, but bear with me.. When you go to school, you go not to learn, but to learn HOW to learn. This takes training for an undisciplined mind. Years of training. When you head off for college and university it's just a progression. The subject matter may be far more complicated, but showing that you have your own personal grasp of an understanding is what "they" are looking for and not just copying text to show you've found the relevant information. So when you do your paper/s, truly come to an understanding on the topic before even starting to write. If you do that, then the answer will flow from you and it'll show anyone that reads that you actually can learn.

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