Basic Examples (2) 

Find the arithmetic derivative of 100:

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Find the arithmetic derivative of the fraction :

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Scope (2) 

Arithmetic derivate for any prime is 1:

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Using Leibniz rule it is easy to find the formula for prime tower :

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which is the same as :

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Properties and Relations (4) 

Higher order arithmetic derivatives are defined successively by nesting the same operation multiple times:

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Arithmetic derivative of an integer n has the following bounds: , where is PrimeOmega[n] and gives the number of prime factors counting multiplicities.

The equality of the lower bound holds if and only if :

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The equality of the upper bound holds if and only if n is prime or a prime tower:

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if and only if where p is any prime:

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It is easy to see that is not a fixed point:

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has nontrivial solution for some rational value x:

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Possible Issues (2) 

The arithmetic derivative does not obey linearity in general:

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Clearly 1 is not same as :

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However, if we find some m and n such that , then for all integer k. For instance:

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Then the following also true:

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Neat Examples (4) 

The graph of for

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We can define the arithmetic integral by solving the equation given a, . The algorithm can be implemented with simple search in the interval found in the aforementioned bounds:

For example, the integrals of 10 and 100:

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We also can find the a's that do not have their integrals less than 1000:

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Here is a list of number with more than one integral:

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The result means that 10 has two integrals, 12 has two integrals and so on.

The Goldbach conjecture implies that the differential equation has a positive integer solution for any natural number :

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For instance as a sum of two primes:

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By the Leibniz rule of arithmetic derivative, 35 is one solution for n such that the differential equation holds:

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Twin prime conjecture implies that the differential equation has infinite number of solutions:

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This is from the application of Leibniz rule on the arithmetic derivative of where p is any prime:

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In general, . If is a prime again,

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Thus we shall have infinite such if the conjecture holds.