$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

Question

A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity

\begin{align*} \zeta(2,1)=\zeta(3)=8\zeta(\overline{2},1) \end{align*}

Such a walk through different methods around a single theme is for me an extraordinary pleasure and a good opportunity to associate and to link aspects which I wasn't aware before.

Here I'm asking for papers like this one which present a single gem and provide us with an anthology of different representions, or different proofs or other aspects around this gem. To keep the selection managable I'd like to put the focus on number theory and combinatorics.

Two more examples which perfectly match my requirements/wishes:

Catalan Addendum by R.P. Stanley provides us with a wealth of different representations of Catalan Numbers extending his $66$ combinatorial representations from the second volume of his classic Enumerative Combinatorics.
The Harmonic Series Diverges Again and Again by S.J. Kifowit and T.A. Stamps presenting some rather elementary proofs together with the follow-up paper More Proofs of Divergence of the Harmonic Series.

Not a paper, but a list: http://www.rzuser.uni-heidelberg.de/~hb3/rchrono.html This is probably the single theorem across mathematics most worthy of collecting different proofs for. — darij grinberg, Aug 15 '15 at 11:48
@darijgrinberg: Great list, many thanks! ... and good characterisation :-) — Markus Scheuer, Aug 15 '15 at 12:23
I'm sure someone has made a collection of proofs of the irrationality of $\sqrt2$. There are quite a few at https://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irrationality — Gerry Myerson, Aug 15 '15 at 13:01

score 3 · Answer 1 · edited Aug 15 '15 at 18:16

3

Fourteen Proofs of a Result About Tiling a Rectangle collects $14$ proofs of the fact that a rectangle tiled by rectangles each of which has at least an integer side has an integer side.

edited Aug 15 '15 at 18:16

rubik

9,344

answered Aug 15 '15 at 12:23

joriki

238,052

score 2 · Answer 2 · answered Aug 15 '15 at 12:54

2

Robin Chapman gives many proofs of $\sum n^{-2}=\pi^2/6$.

answered Aug 15 '15 at 12:54

Gerry Myerson

179,216

score 2 · Answer 3 · edited Apr 13 '17 at 12:58

2

The book "The Fundamental Theorem of Algebra" by Fine and Rosenberger (link) contains detailed discussions of at least six proofs of this theorem, all rooted in different areas of mathematics. Links to other papers (not all in English) compiling various proofs of the theorem can be found at this MathOverflow question.

H. W. Kuhn gave a combinatorial proof in 1974.

edited Apr 13 '17 at 12:58

Community

1

answered Aug 15 '15 at 16:34

Tad

6,679

The link points to good old Springer who kindly is asking only $29.95 / €24.95 / £19.95 for a copy. – marty cohen Aug 15 '15 at 16:58
@marty-cohen I'd be glad to update it to cite an open-source reference if someone finds one. Or come up with my own proof. – Tad Aug 15 '15 at 17:40

score 2 · Accepted Answer · answered Aug 15 '15 at 18:12

2

Elisha S. Loomis, The Pythagorean Proposition, contains $370$ proofs of the Pythagorean theorem. ERIC has a PDF of NCTM reissue of the $1940$ second edition.

answered Aug 15 '15 at 18:12

Brian M. Scott

616,228

Wonderful reference! Thanks a lot! – Markus Scheuer Aug 15 '15 at 18:23

score 1 · Answer 5 · edited Apr 13 '17 at 12:20

1

Here is a question on this site with a whole bunch of proofs of $\sum(n+1)x^n=(1-x)^{-2}$.

edited Apr 13 '17 at 12:20

Community

1

answered Aug 15 '15 at 12:59

Gerry Myerson

179,216

Thanks Gerry! I know about this question, since I've also added an answer to it. :-) But, please let's stick to references to papers in the answer section. – Markus Scheuer Aug 15 '15 at 13:11

$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

5 Answers5