Cody Problem 47843 involved the arithmetic derivative of integers. In particular, 
if p is prime and 
. Therefore, the arithmetic derivatives of 1, 2, 3, 4, 5, and 6 are 0, 1, 1, 4, 1, and 5, respectively.
if p is prime and . Therefore, the arithmetic derivatives of 1, 2, 3, 4, 5, and 6 are 0, 1, 1, 4, 1, and 5, respectively. One might then ask about solving arithmetic differential equations (ADEs). Because the study of differential equations often starts with solving 
, let’s consider the analogous ADE 
. The definition of the arithmetic derivative shows that no prime can solve this equation, but the sample calculations above show that the first (i.e., 
) solution is 4.
, let’s consider the analogous ADE . The definition of the arithmetic derivative shows that no prime can solve this equation, but the sample calculations above show that the first (i.e., ) solution is 4. Write a function to compute the mth solution to this ADE. Because the solutions become large quickly, return the logarithm of the solution.
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I spent quite a while on this before finally realizing the elegant solution.