Diophantine equations with arithmetical functions I want to know is the diophantine equations that contain arithmetic functions are an interesting topic to research? (For example $\varphi(x)=cx-1$ and $\varphi(x)=\sigma(x)-1$.)
$\sigma(x)$ is the sum of divisors of $x$.
 A: This is in response to your question as to whether this is an "interesting or good topic of research." There is no answer to such a question. If you find it interesting, that makes it interesting. If you want to know whether others find it interesting, you can look for (recent) research on such questions. There's a fair amount. I've listed a number of articles below. If you're asking whether this is a major area of current research in number theory, I'd say probably not; but again, to each their own.
Here are some MathStackExchange questions and answers dealing with equations that involve arithmetic functions:

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*Find all $n \in \mathbb N$ such that $\sigma(n) + \phi(n) = n\tau(n)$


*Show that the only solution to $\phi(n) =n-2$ is $n=4$
And here are some articles:

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*Diophantine equations involving Euler's totient function, Yong-Gao Chen, Hao Tian (2017)

*Equations involving arithmetic functions, Gabriel Mincu and
Laurenƫiu Panaitopol, Carpathian Journal of Mathematics. Vol. 22, No.
1/2 (2006), pp. 91-98 (8 pages)

*Luca, Florian, Equations involving arithmetic functions of Fibonacci and Lucas numbers. Fibonacci Quart. 38 (2000), no. 1, 49–55
