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

This is a combinatorics problem and requires checking through the primes for each.

Firstly, the prime numbers we can use are the one below the cubic root of 50,000,000, which are: 2, 3, 5, 7, 11, 13.

Next, we need to find all the combinations of these 6 prime numbers taken 3 at a time, because the problem states that we need to find a number which can be expressed as the sum of a prime square, prime cube, and prime fourth power. The formula for combinations is C(n, r) = n! / r!(n – r)!, where n is the total number of options, r is the number of choices and “!” stands for factorial (the product of all positive integers up to the number). Using the formula, we have 6 options and need to pick 3, thus, C(6,3) = 6! / 3!(6 – 3)! = 20.

Now, we must iterate over these 20 combinations. For each combination, we must check all 6 permutations of the 3 primes (prime one squared, prime two cubed, prime three to the power of four, etc.). After checking all combinations and permutations, we subtract the ones that exceed fifty million.

However, we need to also remember that it’s possible for one number to be represented in different ways. For example, the number 28 from your list can also be represented as 2^2 + 3^3 + 7^4 as well as 2^2 + 3^3 + 5^4, and also as 2^2 + 5^3 + 3^4. This means there may be duplicate counts.

This is a complex problem that cannot be solved by hand as the possibilities are numerous and it would take an extremely long time. Instead, it requires a computer program or algorithm to check the conditions and perform the calculations rapidly.

You could also see this as a problem similar to the Sieve of Eratosthenes which generates primes, but instead of using addition to generate composite numbers, we’re using the addition of various powers of primes. A sievelike computer program could be written based on this concept.

The exact answer to your problem statistically, though, cannot be given without a proper algorithm or computer program to do this extensive computation.

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