Arithmetic Derivative - Inequalities and Bounds

Inequalities and Bounds

E. J. Barbeau examined bounds of the arithmetic derivative. He found that the arithmetic derivative of natural numbers is bounded by

$n' \leq \frac{n \log_k n}{k}$

where k is the least prime in n and

$n' \geq sn^{\frac{s-1}{s}}$

where s is the number of prime factors in n. In both bounds above, equality occurs only if n is a perfect power of 2, that is for some m.

Alexander Loiko, Jonas Olsson and Niklas Dahl found that it is impossible to find similar bounds for the arithmetic derivative extended to rational numbers by proving that between any two rational numbers there are other rationals with arbitrary large or small derivatives.

Read more about this topic: Arithmetic Derivative

Famous quotes containing the words inequalities and/or bounds:

“The only inequalities that matter begin in the mind. It is not income levels but differences in mental equipment that keep people apart, breed feelings of inferiority.”
—Jacquetta Hawkes (b. 1910)

“Firmness yclept in heroes, kings and seamen,
That is, when they succeed; but greatly blamed
As obstinacy, both in men and women,
Whene’er their triumph pales, or star is tamed —
And ‘twill perplex the casuist in morality
To fix the due bounds of this dangerous quality.”
—George Gordon Noel Byron (1788–1824)