With $\varphi(m)$ for $m\ge1$ being Euler's totient function, I am curious to know if are known odd primes of the form $$1+\varphi(m)^\lambda\tag1$$ where $\lambda=2k+1>1$ is an odd integer.
Question. Do you know from the literature if there exists an odd prime with the form in $(1)$? And do you know why it is so difficult to find them? If it was in the literature please answer this question as a reference request and I try to find and read those examples from the literature. Many thanks.
No odd primes of the given form can exist, since $$\varphi(m)^{2k+1}+1=(\varphi(m)+1)(\varphi(m)^{2k}-\varphi(m)^{2k-1}+\dots+1)$$ with $\varphi(m)+1\ge2$ and $\varphi(m)^{2k+1}>\varphi(m)$ for $m>2,k\ge1$. (If $m\le2$ then $\varphi(m)^{2k+1}+1=2$.)