#math trivia for #February12:43 is the smallest number of form p^4+q^3 where p, q are prime. What are p and q? What’s the next number?

— 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

2^{4}+3^{3} = 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 3^{4}+2^{3} and 2^{4}+5^{3}. A quick evaluation tells us that it’s 3^{4}+2^{3}, or 89.

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