Background
In a 2018 question posed by Zhi-Wei Sun, he conjectures that for any rational number $r>0$, there are finite sets $P_r^-$ and $P_r^+$ of primes such that $$r=\sum_{p\in P_r^-}\frac1{p-1}=\sum_{p\in P_r^+}\frac1{p+1}. \tag{1}$$ Recently, Thomas Bloom has answered this question, and has published a preprint on the ArXiv in which a proof is presented that affirms this conjecture.
Prime Egyptian Fractionary Expansions
I'm currently considering a generalization of the representations presented in $(1)$. We could also consider powers of every term in the sum, resulting in a representation of the form: $$q = \sum_{p\in P_q^-}\frac{1}{(p-1)^{a_p}}=\sum_{p\in P_q^+}\frac{1}{(p+1)^{b_p}}. \tag{2} $$
Here, $|P_q^-| = l $ and $ |P_q^+| = m$. Also, $a_p$ and $b_p$ are positive integer numbers for all $p$. Then, the positive and negative Prime Egyptian Fractionairy Expansions$^{*}$ (PEFEs) are defined as $$q_{-} = a_{p_l} \ a_{p_{l-1}} \ \dots \ a_{p_{2}} \ a_{p_{1}}, \tag{3} $$ and $$ q_{+} = b_{p_{m}} \ b_{p_{m-1}} \ \dots \ b_{p_{2}} \ b_{p_{1}}. \tag{4}$$
Example
For instance, we might have the following expansion: $$\frac{3}{8} = \frac{1}{(3-1)^{\color{red}3}} + \frac{1}{(5-1)^{\color{red}1}}. \tag{5} $$
Thus, we obtain the negative PEFE in positional notation: $$ \left( \frac{3}{8} \right)_{-} = \color{red}1 \ \color{red}3 \quad . \tag{6} $$
Also, we have $$\frac{3}{8} = \frac{1}{(3+1)^{\color{blue}1}} + \frac{1}{(7+1)^{\color{blue}1}}. \tag{7}$$
Here, we skipped the prime numbers $2$ and $5$ in the sum. We denote their corresponding $\quad$ powers as $\bar{0}$ in the (in this case, positive) PEFE: $$ \left( \frac{3}{8} \right)_{+} = \color{blue}1 \ \bar{0} \ \color{blue}1 \ \bar{0} \quad. \tag{8} $$
Questions
- Have such prime Egyptian fractionary expansions of the form described in $(2) - (4)$ been described in the literature before?
- Do they bear a relationship with the $p$-adic numbers, and if so, how can this relationship be characterized?
Note
(*) this name is a contraction of Egyptian fraction and binary (or ternary, or quaternary, etc.)