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I am teaching a course on proof. We have learned the methods of proof: direct proof, proof by contrapositive, by contradiction, by induction, etc. We have also done cardinality, modular arithmetic, functions, and a tiny bit of logic. A student approached me and asked if she could do a project on figurate numbers. These are the sequence of numbers you get by increasingly large triangles (triangular numbers), squares (square numbers), pentagons (pentagonal numbers), and so forth. All the exercises I could think of beyond find and prove general formulas for the $n$th number are too elementary. Are there any neat results I could ask her to prove? Thanks.

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There are many difficult questions, not realistically provable by your student, for example, Fermat's polygonal numbrt "theorem." Something that may be accessible is finding triangular numbers that are perfect squares (special Pell equation). –  André Nicolas Nov 26 '12 at 20:51
    
This depends on what background your student has had in your course and earlier courses. Would it be possible for her to discover a closed form for the sum of the first $n$ triangular/square/pentagonal numbers? Could she perhaps look for solutions to a Fermat-like equation $T(a)+T(b)=T(c)$ for triangular numbers? –  Rick Decker Nov 26 '12 at 22:15
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