In this diploma thesis, the combinatorial proofs of Fibonacci and related identities are discussed. The combinatorial proofs prove to be strong tools, useful in different mathematical fields, such as group theory, set theory, analysis and theory of graphs.
Before the proving of identities containing Fibonacci or related numbers, different types of proofs, such as direct algebraic proof, proof with mathematical induction, proof by contradiction and visual proof, are examined and illustrated with cases. Then, Fibonacci numbers are defined and their combinatorial interpretation introduced. The generalisations of Fibonacci numbers are also discussed. A rounded selection of Fibonacci numbers and similar identities are reviewed, and combinatorically proven, in particular with the double counting method. The diploma thesis also discusses some identities that have only recently been combinatorically proven; a combinatorial proof for the Binet’s formula is also cited. Finally, some of the concepts are illustrated also with the help of GeoGebra.
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