derivation of Hartley function
We want to show that the Hartley function log2(n) is the only
function mapping natural numbers to real numbers that
-
1.
H(mn)=H(m)+H(n) (),
-
2.
H(m)≤H(m+1) (monotonicity), and
-
3.
H(2)=1 (normalization).
Let f be a function on positive integers that satisfies the above three properties. Using the additive property, it is easy to see that the value of f(1) must be zero. So we want to show that f(n)=log2(n) for all integers n≥2.
From the additive property, we can show that for any integer n and k,
f(nk)=kf(n). | (1) |
Let a>2 be an integer. Let t be any positive integer. There is a unique integer s determined by
as≤2t<as+1. |
Therefore,
slog2a≤t<(s+1)log2a |
and
st≤1log2a<s+1t. |
On the other hand, by monotonicity,
f(as)≤f(2t)≤f(as+1). |
Using Equation (1) and f(2)=1, we get
sf(a)≤t≤(s+1)f(a), |
and
st≤1f(a)≤s+1t. |
Hence,
|1f(a)-1log2(a)|≤1t. |
Since t can be arbitrarily large, the difference on the left
hand of the above inequality must be zero,
f(a)=log2(a). |
Title | derivation of Hartley function |
---|---|
Canonical name | DerivationOfHartleyFunction |
Date of creation | 2013-03-22 14:32:15 |
Last modified on | 2013-03-22 14:32:15 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 11 |
Author | Mathprof (13753) |
Entry type | Derivation |
Classification | msc 94A17 |