Conjecture $A_1$ : For every $m \in \mathbb N \setminus \{1\}$ there exist mutually different primes $p_{r_1},...,p_{r_m}$ and mutually different primes $b_{w_1},...,b_{w_m}$ and numbers $i_1,...,i_m \in \{1,2\}$ such that $$\displaystyle \sum_{k=1}^m (-1)^{^{i_k}}b_{w_k}p_{r_k}=1$$
Is this known to be true? If not known to be true, I conjecture it is true.
According to your knowledge of the subject, is the number of representations infinite for all $m\geq 2$?
This I would like to formulate as:
Conjecture $B$ : The number of representations is infinite for all $m \geq 2$
And, with the help and computational efforts of Peter it seems that the conjecture $A_1$ can be strengthened to:
Conjecture $A_2$: For every $m \in \mathbb N \setminus \{1\}$ there exist mutually different primes $p_{r_1},...,p_{r_m}$ and mutually different primes $b_{w_1},...,b_{w_m}$ such that $$\displaystyle p_{r_1}b_{w_1-} \sum_{k=2}^m b_{w_k}p_{r_k}=1$$
Some observations and computations even suggest that in conjecture $A_1$ and $A_2$ there are examples where all the mutually different primes $p_{r_1},...,p_{r_m}$ also can be mutually different from all the mutually different primes $b_{w_1},...,b_{w_m}$.
What is already known about these and similar problems?