# On prime conjectures and the abc conjecture

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 both examples I don't know what about the behaviour of the corresponding sequences $a_n$, thus I don't know is such idea to combine a conjecture about primes and the abc conjecture in this way is useful. That is what I am asking in my Question: Is this strategy potentially useful with the purpose to deduce some interesting statement about the distribution of prime numbers? Also I don't know if is feasible to disprove the veracity of some conjecture about prime numbers if one presume that the abc conjecture holds: is feasible currently this way or basically is impossible? – user243301 Apr 24 '17 at 8:06
• @Alex many thanks for your help editing the question. – user243301 Apr 24 '17 at 18:49
• With this BOUNTY I am interested to know if it's possible to combine the abc conjecture with this kind of prime conjectures with the purpose to study the veracity of such conjectures (that is we presume that the abc conjecture is true y we combine it with prime conjectures, if it is feasible, to deduce some interesting). Feel free to criticize my strategy. Notice that since I presume that the abc conjecture is true, my only purpose is to know if this strategy is potentially interesing versus prime conjectures. Many thanks. – user243301 Apr 29 '17 at 9:30
• @AlexFrancisco many thanks for previous edit. – user243301 Feb 10 '18 at 8:39