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Fibonacci in Hogwarts?

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Abstract

An elementary algebraic problem attributed to Leonardo of Pisa is analysed and some illogical elements in its formulation and solution are exposed. The natural context in which the problem was formulated is then proposed, and some consequences are discussed. How many times have you heard that somebody is a wizard? And how many times did you take it literally? Most likely, the answer to the second question is ‘never’. And yet, there are reasons to believe that some people among us are real wizards, of the kind described with so much charm in the recently published series of books on Harry Potter [ 1, 2, 3, 4 ]. These books have provided us with a wealth of details on wizards and their society.

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In [1], Tomislav Došlić studies a special case of the classic Ass and Mule Problem and determines when integral data give integral solutions. The question of when integral data gives integral solutions in the general case intrigued me some years ago; I even published the assertion that I could not see any simpler result than computing the answers and seeing if the results were integral [2]. However, I later found a satisfactory result for this and some variants of the problem, which are given in [3, 4]. Here I give this result, which describes an infinite set, and show it leads to a simpler derivation of Došlić's interesting result that his variation has only 16 positive integral solutions. A different method permits finding the nonpositive integral solutions. I will then state a similar but unsolved problem.
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