Definition

$bulletquad\; p\text{'}\; =\; 1text\{\; for\; any\; prime\; \}p$

$bulletquad\; (ab)\text{'}=a\text{'}b+ab\text{'}text\{\; for\; any\; \}a,b\; in\; mathbb\{N\}qquad\{\}$ (Leibnitz rule).

To coincide with the Leibnitz rule 1′ is defined to be 0, as is 0′. Explicitly, assume that

$x\; =\; p\_1^\{e\_1\}cdots\; p\_k^\{e\_k\},$

where p₁, ..., p_k are distinct primes and e₁, ..., e_k are positive integers. Then

$x\text{'}\; =\; sum\_\{i=1\}^k\; e\_ip\_1^\{e\_1\}cdots\; p\_i^\{e\_i-1\}cdots\; p\_k^\{e\_k\}\; =\; sum\_\{i=1\}^k\; frac\{e\_i\}\{p\_i\}\; x.$

The arithmetic derivative also preserves the power rule (for primes):

$(p^a)\text{'}\; =\; ap^\{a-1\},$

where p is prime and a is a positive integer. For example,

$81\text{'}\; =\; (3^4)\text{'}\; =\; (9*9)\text{'}\; =\; 9\text{'}*9\; +\; 9*9\text{'}\; =\; 2[9*(3*3)\text{'}]\; =\; 2[9*(3\text{'}*3\; +\; 3*3\text{'})]\; =\; 2[9*6]\; =\; 108\; =\; 4*3^3.$

The sequence of number derivatives for k = 0, 1, 2, ... begins :

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, ...

E.J. Barbeau was the first to formalize this definition. He extended it to all integers by proving that $(-x)\text{'}\; =\; -x\text{'}$ uniquely defines the derivative over the integers. Barbeau also further extended it to rational numbers. Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents e_i are allowed to be arbitrary rational numbers.

Relevance to number theory

Ufnarovski and Åhlander have detailed the function's connection to famous number-theoretic conjectures like the twin prime conjecture, the prime triples conjecture, and Goldbach's conjecture. For example, Goldbach's conjecture would imply the existence of an n so that n' = 2k for every k, and the twin prime conjecture would imply that there are infinitely many k for which k'' = 1.

See also

• Field with one element

References

• E. J. Barbeau, "Remark on an arithmetic derivative", Canadian Mathematical Bulletin Vol. 4 (1961), 117–122.
• Victor Ufnarovski and Bo Åhlander, "How to Differentiate a Number", Journal of Integer Sequences Vol. 6 (2003), Article 03.3.4.
• Arithmetic Derivative, Planet Math, accessed 04:15, 9 April 2008 (UTC)
• L. Westrick. Investigations of the Number Derivative.
• Peterson, I. Math Trek: Deriving the Structure of Numbers.