On Upper Bounds with $ABC=2^m p^n$ and $ABC= 2^m p^n q^r$ with $p$ and $q$ as Mersenne or Fermat Primes (1809.03328v1)

Abstract: Per the $ABC$ conjecture, for every $\varepsilon > 0$, there exist finitely many triples ${A, B, C}$, satisfying $A + B = C, gcd(A,B) = 1, B > A \geq 1$, such that $C > rad(ABC)^{1 + \varepsilon}$, where $rad(ABC)$ is the product of all distinct primes constituting $ABC$. It is shown (possibly known) that for any $\varepsilon > 0$, $rad(ABC)^{1 + \varepsilon} > C$ holds for $ABC$ of the form $2^mp^n$ such that $2^m + \mu = p^n$, where $m, n$ are positive integers, $\mu = \pm 1$, and $p$ is an odd prime. Condition $2^m + \mu = p^n$ requires $p$ to be either a Mersenne or Fermat prime. For $ABC = 2^mp^nq^r$, $p$ and $q$ either Fermat or Mersenne primes but distinct, and $m,n,r$ positive integers, such that any diophantine relation, $\mbox{(i)}\; 2^m + \mu = p^nq^r, \mbox{(ii)} \; 2^mp^n + \mu = q^r \mbox{ or } \mbox{(iii)} \; p^n + \mu q^r = 2^m$, holds, using results in literature and mostly elementary methods, the number of triples ${A, B, C }$ are shown to be finite, except for ${2^{y+1}, 2^{2y}+1, (2^y+1)^2}$, a solution to (iii) with $y$ such that $2^y+1$ and $2^{2y}+1$ are primes, in which case $rad(ABC)^{1 + \varepsilon} > C$ holds for any $\varepsilon > 0$.