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Is it appropriate to use conjectures in math contest?

I've been to fair amount of math contest and as far as I know the judges want a solid proof for every step take to make to solve the problem. But is using a conjecture a good idea? Will they accept the answer or will they deduct points?

Conjectures are statements that are based on an incomplete information and are unproven. But many famous conjectures can solve a contest problem.

Let's imagine we are given an equation with 3 variables we need to find solutions for it in prime numbers. We somehow come up to this equation:

$A^{11} + B^7 = C^{13}$

At this point we could use the Beal's conjecture to prove that $A,B \text{ and } C$ have to have a common factor in order to have a solution. But this means that they are not prime numbers, leading to that there isn't prime solution for that equation.

But until now the Beal's conjecture isn't proven. Should I use it?

Also there are lot of well-know conjectures like both Goldbach conjecture, Collatz conjecture, Fermat-Catalan conjecture...

P.S. I don't want a solution with proof for the equation I just think of it.

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  • $\begingroup$ Isn't it that in math contests there is alwaays an (more or less) easy answer to the problem ? $\endgroup$ – Dominic Michaelis Sep 9 '13 at 19:42
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    $\begingroup$ Short answer, no conjectures are tolerated in any math contest I know of. And that's perfectly fine if you ask me. @DominicMichaelis: depends on the contest........ $\endgroup$ – darij grinberg Sep 9 '13 at 19:43
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    $\begingroup$ No. Conceivably one might get $1$ out of $7$ if the grader is in an exceptionally good mood, and the connection is non-trivial. $\endgroup$ – André Nicolas Sep 9 '13 at 19:43
  • $\begingroup$ @DominicMichaelis Yes always there exist a proven solution, otherwise they won't give you that problem, but you can't come to solution and you know that conjecture can be used, but is it acceptable? $\endgroup$ – Stefan4024 Sep 9 '13 at 19:45
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No, it is not appropriate to use conjectures to do a question in any written exam or test or contest unless it is explicitly stated otherwise. Doing mathematics is all about proofs, and conjectures are statements which have neither been proven nor disproven.

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    $\begingroup$ + it hasn't been proven that you can't prove or disprove it $\endgroup$ – Dominic Michaelis Sep 9 '13 at 19:45
  • $\begingroup$ That would have been my conjecture as well $\endgroup$ – Hagen von Eitzen Sep 9 '13 at 20:03
  • $\begingroup$ I completely agree with you about contest questions. Grading of exams can be more elastic. $\endgroup$ – André Nicolas Sep 9 '13 at 20:03

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