An Introduction to the Language of Mathematics by Frédéric Mynard

Author:Frédéric Mynard
Language: eng
Format: epub, pdf
ISBN: 9783030006419
Publisher: Springer International Publishing

27Indeed, as by definition

we have

28That is, d is a common divisor of a and b and additionally,

29This completes the base case of the induction.

30See, e.g., the wikipedia entry on “Nim” for more details.

31Note that we do not really need to check the case n = 2, but this is safer in the context where we use the definition of F n via (2.32), which is inductive of depth 2, that is, a given F n depends on the previous 2 terms of the sequence. Sometimes in using this definition, we really assume more that P(n), namely both P(n) and P(n − 1), in deducing P(n + 1). In such a case, we need to include P(1) and P(2) in the base case. To be on the safe side, we’ll check for the first two values of n whenever we prove something on Fibonacci numbers.

If a term depended on the previous 3 terms, then it would be safer to check P(1), P(2), and P(3) for the base case, for you may not realize that you are assuming more than P(n) to get P(n + 1).

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