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Importance of the Absolute ValueProfessor Ducci's observation pertains to the problem of integer iterations on a circle:
Whatever 4 integers you start with, you'll reach the sequence of four zeros after a (very) finite number of iterations. So it may appear surprising that with the removal of the absolute value, the terms of the sequence grow without bound. 
Copyright © 1996-2010 Alexander Bogomolny
Let a, b, c, d be the numbers originally placed on the circle. These are replaced with
Let One way of thinking of the magnitude of the four numbers is to consider them as components of a 4-dimensional vector. Indeed, the norm of such a vector grows without bound only if the same happens to at least one of its components; for the norm measures the distance of the vector's end point to the origin. We'll use the Euclidean norm:
On the other hand,
Adding the two gives
meaning that the square of the norm at least doubles with each iteration. Reference
Copyright © 1996-2010 Alexander Bogomolny |
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