## Prime power triples

### Problem 87

Published on Friday, 21st January 2005, 06:00 pm; Solved by 12315; Difficulty rating: 20%The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is 28. In fact, there are exactly four numbers below fifty that can be expressed in such a way:

28 = 2^{2} + 2^{3} + 2^{4}

33 = 3^{2} + 2^{3} + 2^{4}

49 = 5^{2} + 2^{3} + 2^{4}

47 = 2^{2} + 3^{3} + 2^{4}

How many numbers below fifty million can be expressed as the sum of a prime square, prime cube, and prime fourth power?