I have had this conjecture for a while:
Let $q\in\mathbb R\setminus [-1,1]$ and let $p_n$ be the $n^\text{th}$ prime. Then: $$\sum_{n=1}^{\infty}\bigg(\frac 1{q^{p_n}-1}-\sum_{k=1}^{n-1}\frac 1{q^{p_n p_{n-k}}-1}\bigg)=\frac 1{q(q-1)}$$
The sum looks like this when we expand it out:
$$\frac 1{...
I have had this conjecture for a while:
Let $q\in\mathbb R\setminus [-1,1]$ and let $p_n$ be the $n^\text{th}$ prime. Then: $$\sum_{n=1}^{\infty}\bigg(\frac 1{q^{p_n}-1}-\sum_{k=1}^{n-1}\frac 1{q^{p_n p_{n-k}}-1}\bigg)=\frac 1{q(q-1)}$$
The sum looks like this when we expand it out:
$$\frac 1{...
