let $p_1$ and $p_2$ be positive primes such that $p_1,p_2 \equiv 3\bmod 4$ and $\phi$ is the Euler totiont function , I want to find the Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ ? I can't find such pairs satisfying the titled claim , I have used the fact that : $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod \frac{{p_1}-3}{p_2}$ but this dosn't give any thing to determine all pairs $(p_1,p_2)$ for which the titled claim w'd be satisfied .

**Edit** I have Edited the question to avoid any complication and working with ordinary prime which are $3$ modulo $4$ in the same time gives the definition of Gaussian primes

**Note**:The motivation of this question is to know more about behavior of Euler totiont function with Gaussian primes