# How important is it to come up with or learn an elementary solution?

Note: by "elementary" I mean "without using more advanced theory and tools".

Students are sometimes required or encouraged to solve very difficult problems using limited number of tools and machinery. It is not uncommon in competition-style exams that problems in algebraic/analytic number theory are solved using elementary number theory (i.e. without analysis or abstract algebra). Various other exams in the world also have a limit on the number of tools that can be used. For example, sometimes, people are required to prove something similar to mean value theorem for a specific given function (with an explicit expression) solely by very complicated algebraic computations, without Calculus, because the rigorous $$\epsilon$$-$$\delta$$ definition of derivatives and limits are not yet taught.

What are the reasons why we wish to tackle hard problems with elementary methods? And is it beneficial or not?

If we are building a theory from axioms, when we write proofs, we must only use things that are already proven. But in the situation I describe above, we are not building up a theory; instead, we are applying some theory to solve a problem, so in this situation, what are the reasons why we sometimes limit our range of tools? What are some pros/cons of this?

This might be, to some extent, a matter of taste, but it is still interesting to know reasons for this.

• This sort of feels like another question where there's an unstated assumption that the goals of IMO competitions should be the same as the goals of advanced math education. Who cares if someone wants to run a competition like IMO? How's it hurting you? Should we outlaw marathon races because the practical way to travel 26 miles is by bicycle (or car)? Note also that this question is amazingly similar to your previous question: matheducators.stackexchange.com/questions/16858/… – guest Oct 22 '20 at 14:23
• In the case of students, this can be like weight-training for football players, and for researchers, see my comment that begins with as I began to realize how one can sometimes. – Dave L Renfro Oct 22 '20 at 16:46
• I just don't think using too low-level tools is beneficial to solving a problem because it's like reinventing the wheel or doing much strenuous effort than necessary. eg there's nothing wrong with using all of the course of Algebra to prove FTA with Galois theory. There's an easier topological proof too. FTA cannot be proven with just high school algebra - it can but it takes much more effort, handwaving and sacrificing some rigor? – Lenny Oct 22 '20 at 17:59
• Regarding low level tools, it's well known that you can get all sorts of information from hard analysis proofs in analysis (e.g. explicit $\epsilon$-$\delta$ error estimates and the like that typically don't require highly sophisticated ideas) that you can't get from soft analysis proofs in analysis (e.g. Baire category existence results), and vice-versa. See here and here for these terms. – Dave L Renfro Oct 22 '20 at 20:23
• Perhaps similar to this. In the Boston Marathon, you are required to travel on foot,you are not allowed to use bicycles or motorcycles. – Gerald Edgar Nov 16 '20 at 11:36