Arithmetic derivative

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In mathematics, specifically number theory, the arithmetic derivative is an analogue of the derivative used in mathematical analysis.

1 Definition
2 Relevance to number theory
3 See also
4 References

[edit] Definition

For natural numbers defined as follows:

$p' = 1$ for any prime $p$
$(a b)' = a' b + a b'$ for any $a,b \in \mathbb{N}$ (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

$x' = \sum_{i=1}^k e_ip_1^{e_1}\cdots p_i^{e_i-1}\cdots p_k^{e_k} = \sum_{i=1}^k \frac{e_i}{p_i}x$ .

E.J. Barbeau was the first to formalize this definition. He extended it to all integers by proving that $( - x)' = - x'$ uniquely defines the derivative over the integers. Barbeau also further extended it 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.

[edit] 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.

[edit] See also

Field with one element

[edit] 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)
Deriving the Structure of Numbers, Science News Online, accessed 04:15, 9 April 2008 (UTC)