About semiprimal representations of $1$ 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? 
 A: My guess is that at least for $m$ sufficiently large, you can deduce your conjectures from the following paper https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/quadratic-form-in-nine-prime-variables/5B4875E6EB48D285736D7672419F8BF7 or other related work. 
Let us fix $i_1, ..., i_m \in \{1,2\}$ not all $2$. Let us forget about the mutually distinct condition for a second. You can form a quadratic form
$$
f(x_1, ..., x_m, y_1, ..., y_m) = \sum_{j=1}^m (-1)^{i_j}x_j y_j. 
$$
Then you can obtain an asymptotic formula for the number of prime solutions to the equation $f(x_1, ..., x_m, y_1, ..., y_m)  = 1$ by the mentioned paper, provided that the conditions of the theorem is satisfied, which I expect to be the case for $m$ large enough. 
To obtain the mutually distinct condition one can remove the number of solutions where two of them are equal, for example one case to consider would be the number of prime solutions of $f(x_1, x_1, x_3 ..., x_m, y_1, ..., y_m)  = 1$, which again can be computed using the above paper. 
Also another thing perhaps you can try is pick any three primes, say $2,3$ and $5$. 
Consider the linear equation 
$$
2x - 3y - 5z = 1. 
$$
You can probably prove that there are infinitely many prime solutions to this equation by modifying the proof of ternary Goldbach or by applying https://annals.math.princeton.edu/2010/171-3/p08. 
