#math trivia #43 solution

#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

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.

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