<s>[INST] Prime number of the form $A^{B^C}+D^{E^F}$ My Brother asked me what is the smallest prime number of the form $A^{B^C}+D^{E^F}$ where A,B,C are three distinct prime numbers, and D,E,F are 3 distinct primes that is Permutations of those 3 primes.I realize that we must arrange the exponents to be as smallest as possible in order to get smallest prime as possible.So what is the smallest such prime? (and second smallest :-)
 [/INST] $A = 2$ OR $D = 2$
$A \neq D$, $B \neq E$, $C \neq F$
$a^n$ generally increases faster as $n \rightarrow \infty$ than as $a$ does.
If we say that $A = 2$, then $A^{B^C}$ must be even. Therefore, $D^{E^F}$ must be odd.
Therefore we have $2^{B^C} + D^{2^F}$ OR $2^{B^C} + D^{F^2}$
Let's try $2^{5^3}+3^{2^5}$. It is prime! Are there any smaller primes of the specified form? Taking $2^{B^C} + D^{2^F}$, we have two cases: is $5^3 < 3^5$ and is $2^5 < 5^2$. The first cases is true, but the second case is not. Therefore, let's try varying the second case. Is $2^{5^3}+3^{5^2}$ prime? It is not! Therefore, $2^{5^3}+3^{2^5}$ is the answer!
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