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Per the $ABC$ conjecture, for every $\varepsilon > 0$, there exist finitelymany 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 ofall distinct primes constituting $ABC$. It is shown (possibly known) that forany $\varepsilon > 0$, $rad(ABC)^{1 + \varepsilon} > C$ holds for $ABC$ of theform $2^mp^n$ 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, thenumber 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 forany $\varepsilon > 0$.

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