Abstract
In Part I of this article we considered some binomial identities and also some identities involving the Fibonacci numbers, and proved them using methods which we described as ‘largely’ combinatorial. Now we shift our focus to number theory and to prime numbers in particular, and showcase some proofs having a strong combinatorial element. Throughout this article, p denotes an odd prime.
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M Aigner and G M Ziegler, Proofs From The Book, Springer Verlag, 2001. This book is highly recommended; it has a rich collection of proofs of all kinds — some of the most famous and beautiful proofs in mathematics. We have taken Erdös’s proof of the divergence \(\sum {\tfrac{1} {p}}\) of from this book.
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Shailesh Shirali is Director of Sahyadri School (KFI), Pune, and also Head of the Community Mathematics Centre in Rishi Valley School (AP). He has been in the field of mathematics education for three decades, and has been closely involved with the Math Olympiad movement in India. He is the author of many mathematics books addressed to high school students, and serves as an editor for Resonance and for At Right Angles, He is engaged in many outreach projects in teacher education.
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Shirali, S.A. Combinatorial proofs and algebraic proofs — II. Reson 18, 738–747 (2013). https://doi.org/10.1007/s12045-013-0095-2
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DOI: https://doi.org/10.1007/s12045-013-0095-2