The sum of two prime powers equal a third prime power.

Question

Is

$$13^2 + 7^3 = 512=2^9$$

the only solution for the sum of two primes $p,q$ raised to powers greater than $1$ equals a third prime power?

Answer 1

No, it isn't. For instance, we have:

• $2^n+2^n=2^{n+1}$, for all $n$ meeting problem constraints; and
• $2^4+3^2=5^2$.

Answer 2

You can check these types of numbers on wolfram alpha.

However you can see that $3^1+6^1=9^1$