h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hi there,

is there a relation to the Euler function [f(x) = e^x], besides giving the original function, when differentiated? Is it possible to calculate Fibonacci numbers with it? Is it true that e^x belongs to the "keep on doubling it" type?

HP »--.

e^x is a continuous function, so differentiating it isn't the same as taking the differentials of a series. If you take the series of numbers x[n]=e^n, where n=0,1,2,3,... then the differentials are not the original series. The keep on doubling it are the type x[n]=a2^n. a=any number.

Can I have a quick moan about the picture caption? OK, it refers to prime numbers, which is *wrong*.

I agree with your moan! And thanks for your explanation!

HP »--.

That is me being a complete brain donor. I'll change it now...

Graphic looks great now! Thanks!

Wow, my explanation made sense and hasn't been pulled apart by fellow mathematicians yet.

But dear Bagpuss, don't question your abilities!

However, you are talking about a misguiding statement in the article. First I don't know what 'differential' means. Well, maybe I can imagine a definition. But then it says that 'The first differential becomes the same as the series itself'. This is inaccurate, because it is shifted by one position (which is not true for 2^n).

If shifting is allowed, the article gets another problem: The Cefpret numbers 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41,... have the same property, although, according to the article, Fibonacci and 2^n are the only ones.

By the way, there is a (not too complicated) connection between the discrete realm of such series and the continuous realm of linear differential equations. (I once had to find the formulae of the article as a homework.)

A very nice article, especially because it has the perfect level of complexity for the Guide.

I agree the article is misleading in its use of the word 'differential'. What the article should have discussed were 'differences' (as made famous by Charles Babbage and his Difference Engine). Differences can be used to determine the order of a polynomial. I believe that the article is also slightly misleading when it states that the simplest series starts 1,1,2... I was always under the impression that it started 0,1,1,2... thus avoiding the obvious question 'Why start with two ones?'

The irrational number (phi) at the heart of this series is also worth studying in its own right. It is the number of Golden Section, known to the ancient Greeks, among others. It approximates to 1.618034. Its reciprocal (1/phi) is 0.618034(approx) and its square is 2.618034(approx).

Yes - now you mention it - "differences" would have been better ...and I like the idea of starting from 0 too!

But surely that begs the question "Why should the second number be a 1?"

Hehehe. ...."because"!

Silly me...