Anyone who has stayed in math classes long enough to reach calculus quickly comes to believe that calculus is more advanced and complex than arithmetic. And while that may be true for the most intuitive aspects of arithmetic that we learn in grade school, the seemingly innocent discipline quickly becomes more mysterious as one advances into it.
Consider the most basic operation in calculus, the derivative. The derivative of the a function is said to describe the instantaneous rate of change of function (how fast it is going up or down in any given direction), or alternatively the slope of a function at any given point. However, there is a related operation on integers is called the arithmetic derivative, defined as follows:
 p’ = 1 for any prime number p
 (ab)’ = a’b + ab’ for any a,b ∈ ℕ
 1′ = 0
 0′ = 0
While there is an intuitive and even “visual” nature to the calculus derivative, the arithmetic derivative seems more abstract and opaque. Additionally, the former is about continuous functions, while the latter is about discrete entities such as integers or rational numbers. The main thing the two concepts have in common is that they obey the Liebniz Rule governing the products of derivatives, as described on line 2 above.
So can the arithmetic derivative tell us anything useful about integers? It is intimately tied to prime numbers and prime factorization, so it is in that sense an additional tool for examining fundamental properties of numbers as they relate to primes and patterns of primes. The article Deriving the Structure of Numbers quotes Linda Westrick, who has studied the arithmetic derivative and says that it “provides a different context from which to view many topics of number theory, especially those concerning prime numbers. The complex patterns which arise from its simple definition make it interesting and worthy of study.” To see such patterns, one can apply the derivative repeatedly, n’, (n’)’, etc., just as one would in calculus, to chart the variations even among the first few natural numbers.
n

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

n‘

0

1

1

4

1

5

1

12

6

7

1

16

1

9

8

32

1

21

n"

0

0

0

4

0

1

0

16

5

1

0

32

0

6

12

80

0

10

n"’

0

0

0

4

0

0

0

32

1

0

0

80

0

5

16

176

0

7

Indeed the interesting patterns include the fact that some numbers quickly go to zero as one repeatedly applies the arithmetic derivative, as on the case of 6′ = 5, 6” = 1, 6”’ = 0. Other numbers, seem to increase without bound. In the above chart, for example, powers of 2 appear to grow quite quickly. Yet others bounce around unpredictably somewhere in between. In many ways, this reminds me a bit of the Collatz Conjecture which have discussed on CatSynth in the past.
Perhaps the most intriguing property of the arithmetic derivative is its relation to twin primes, pairs of prime numbers whose difference is 2, like 11 and 13 or 29 and 31. It is conjectured that there are infinitely many such pairs of twin primes, but no one has ever proven this. However, it turns out that twin primes are related to the second derivative of a number n”. So if there are infinitely many numbers n for which n” = 1, then there are infinitely many twin primes. As described in this paper by Victor Ufnarovsk, one can derive this from the following theorem:
Let p be a prime and a = p+2. Then 2p is a solution for the equation n’ = a
The proof is relatively straightforward:
(2p)’ = 2’p + 2p’ = p + 2
So if 2p’ = p + 2 and p + 2 is also prime, as would be the case for twin primes, then 2p” = (p + 2)’ = 1.
Unfortunately, this does not answer the question of whether there are infinitely many such pairs. So the famous problem remains open.