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arithmetic derivative

Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

11Z05 no label found

Comments

While browsing the encyclopaedia, I came across the definition of derivative of a number. It is certainly an inteteresting illustration of how Leibniz's rule can be used to define a notion of derivative in a context where one would ordinarily not think of differential calculus. My question to you is "What is this notion of integer derivative good for?", meaning "Are you aware of any theorems of number theory which can be proven using the integer derivative?".

... when the derivative is zero, the number is a
constant :-)

Matte

What exactly do you mean by this??? I would like to think that every single integer is a constant.

Johan

Of course every integer is a constant.
My comment was just a joke.
Nevertheless, I'm too curious how
this integer derivative is used.

Matte

Apparently Goldbach's conjecture can be stated using
the number derivative as:

if a\in Z then there exists a b\in Z such that
b'=2a.

This is mentioned in
http://web.mit.edu/lwest/www/intmain.pdf
http://www.maa.org/mathland/mathtrek_03_22_04.html

The number derivative is also related to the
twin prime conjecture:

http://www.lacim.uqam.ca/~plouffe/JIS/ufnarovski.pdf

Matte

It's unfortunate that this number derivative is not compatible with addition. In general (n+m)' != n' + m'. If it were, it would have made a nice example of a differential ring (field) if extended to the integers (rationals).

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