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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 $x$.

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    $\begingroup$ These are not typically called Diophantine equations, which generally refer to systems of pol\iynomial equations. But equations of the sort you're asking abouot are often studied. For example, the equation $\sigma(n)=2n$ characterizes perfect numbers. And there are also equations involving the Euler $\phi$ function that have been much studied. $\endgroup$
    – Joe Silverman
    Commented May 16, 2022 at 20:50
  • $\begingroup$ Is this topic not good as topic of research ? $\endgroup$
    – user482376
    Commented May 16, 2022 at 21:33
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    $\begingroup$ your examples, especially the second one, look strange $\endgroup$
    – Fedor Petrov
    Commented May 17, 2022 at 5:54
  • $\begingroup$ $\sigma(x)$ is the sum of divisors of $x$. $\endgroup$
    – user482376
    Commented May 17, 2022 at 6:25
  • $\begingroup$ This seems to be quite close to your other question: Diophantine equations or associative operations on ordered lattice. $\endgroup$
    – Martin Sleziak
    Commented May 17, 2022 at 6:26

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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:

And here are some articles:

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