— Burt Kaliski Jr. (@modulomathy) February 12, 2012
This is a quick one. If 43 is the smallest number of the given form, then p, having the larger exponent, must be the smallest prime, so q must be the second smallest prime. The solution is thus p = 2, q = 3, which can be checked as
24+33 = 16 + 27 = 43.
Note that in problems like this, the use of different symbols for the primes generally means one may assume the primes are distinct. In any case, p = q = 2 would yield 24 as the “smallest” of form.
The “next number” — meaning the “next smallest number of the same form” — must be the smaller of 34+23 and 24+53. A quick evaluation tells us that it’s 34+23, or 89.