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Problem 87

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Prime Power Triples

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:

$$\begin{array}{rl}28& =2^2+2^3+2^4\\ 33& =3^2+2^3+2^4\\ 49& =5^2+2^3+2^4\\ 47& =2^2+3^3+2^4\end{array}$$

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

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素数幂三元组

最小的、可以表示为一个素数的平方、一个素数的立方和一个素数的四次方之和的数是$28$。实际上，在小于$50$的数中，一共有$4$个数满足这一性质：

$$\begin{array}{rl}28& =2^2+2^3+2^4\\ 33& =3^2+2^3+2^4\\ 49& =5^2+2^3+2^4\\ 47& =2^2+3^3+2^4\end{array}$$

在小于五千万的数中，有多少个数满足上述性质？

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