<s>[INST] Is $2^n$ always a sum of exactly $n$ primes for $n \geq 2$? For example:
$2^2=4=2+2$
$2^3=8=3+3+2$
$2^4=16=5+5+3+3$
$2^5=32=17+7+3+3+2$
$2^6=64=17+17+17+7+3+3$
How much further can this go?
 [/INST] This is a very difficult problem but luckily previous results allow us to solve it (thanks to @Luke_Collins). The Weak Goldbach Conjecture states that every odd number $n$ (greater than or equal to $7$) is equal to the sum of three prime numbers. This implies that every even number greater than or equal to $10$ is equal to the sum of $4$ primes. Then consider $n\geq 4$
$$2^n=[2^n-2(n-4)]+2(n-4)$$
We can write $2(n-4)=2+2+\dots+2$ as the sum of $n-4$ $2$'s. Since $2^n-2(n-4)\geq 10$ (and even) for all $n\geq 4$, we are assured that it can be written as the sum of $4$ primes. Since the cases $n\in\{1,2,3\}$ are easy to check, every power $2^n$ can be written as the sum of $n$ primes.
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