We denote the $k$-th prime number number with $p_k$ for $k\geq 1$.
Are well known some conjectures about primes, you can a very good summary is this Conjecture 78 from Carlos Rivera website The prime puzzles & problems connection.
Sometimes when I know a new conjecture from this or a different source I do the following exercise illustrated in this example
Example 1. Let the Firoozbakht's conjecture, see it in this Wikipedia, then exponentiating such inequality I can define a sequence (OEIS A182519) of integers $a(n):=a_n$, for $n>1$, such that $$a_n+p_{n+1}^n=p_{n}^{n+1}$$ and each of such triples $(a(n),p_{n+1}^n,p_{n}^{n+1})$ satisfies $\gcd(a(n),p_{n+1}^n,p_{n}^{n+1})=1$. Thus I can to combine with the abc conjecture, I say the version abc Conjecture II from this Wikipedia, to state that $\forall ε>0$ there exists a constant $K_ε$ such that for all such triples ($n>1$) $$p_{n}^{n+1}\leq K_ε\cdot (p_n p_{n+1})^{1+ε}\left(\operatorname{rad}(a_n)\right)^{1+ε}.$$
Example 2. Similarly one can do calculations for Reza's conjecture (Reza Farhadian, Lorestan Univesity) and the abc Conjecture to conclude that $\forall ε>0$ there exists a constant $C_ε$ such that for all triples that one can defines easily from Reza's inequality ($n>4$) $$n^{p_n^{n+1}}=\left(n^{p_n}\right)^{p_n^n}\leq C_ε\cdot \left(\operatorname{rad}(np_n a_n )\right)^{1+ε}$$ (Here a mistake was fixed.) when we presume (I don't know if the following hypothesis is superfluous, I know that $\gcd(n,p_n)=1$)) $$\gcd(a(n),n,p_n)=1.$$
Question. I would like to know from a divulgative viewpoint, that is heuristics or your ideas, reasonings... about if such idea to combine a conjecture about primes of this kind and the abc conjecture is potentially useful to deduce interesting statements about primes. Why should be useful to combine the abc conjecture with this kind of conjectures of the cited professors? Why no? Do you know an example in which an author does combine the abc conjecture and a conjecture of this kind to deduce some interesting statement (then cite the reference)? Many thanks.
In 2017, R. Farhadian and R. Jakimczuk proved that the Farhadian's conjecture (or Reza's conjecture) is true for almost all prime numbers. For their work you can see
*Farhadian, R., & Jakimczuk, R. (2017). On a new conjecture of prime numbers. International Mathematical Forum, 12, 559-564. doi:10.12988/imf.2017.7335.
This new result may be useful for studying the abc conjecture.