Inspired by Is $29$ the only prime of the form $p^p+2$?.
Claims on prime powers and their alternating sums
Consider the expressions $\mathcal P=\sum\limits_{i=1}^n p_i^{p_i}$ and $\mathcal Q=\sum\limits_{i=1}^n (-1)^ip_i^{p_i}$ over the prime numbers $(p_i)_{i\in\Bbb N}$.
Then I claim that
$n=2,4,24$ are the only occasions when $\mathcal P$ is prime.
$n=2,4$ are the only occasions when $\mathcal Q$ is prime.
PARI/GP code is for(n=1,+oo,if(isprime(sum(i=1,2*n,prime(i)^prime(i)))==1,print(n))). Note that we need only check even $n$ as all prime powers are odd excluding $p_1=2$.
Frankly I have no clue where to start. However, $\mathcal P$ and $\mathcal Q$ can be represented as $$\pm2^2+3^3+x\,\text{terms of the form}\,(6k-1)^{6k-1}+(n-x-2)\,\text{terms of the form}\,(6k+1)^{6k+1}.$$ Now working in modulo $6$, $$\mathcal P\equiv 4+0+x\cdot(-1)^{-1}+(n-x-2)\cdot1^1\equiv2-2x+n\pmod6$$ so $\mathcal P$ is always composite when $n\equiv2(x-1)\pmod6$. Similarly, $$\mathcal Q\equiv -4+0+x\cdot(-1)^{-1}+(n-x-2)\cdot1^1\equiv n-2x\pmod6$$ so $\mathcal Q$ is always composite when $n\equiv2x\pmod6$. Despite this, it is difficult to verify these congruences since $x$ can be hard to determine when $n$ is large.
Nonetheless, I suspect a modular approach like the one above is too simple to solve the problem. I would appreciate any advances on this, though I would like further values of $n$ through computation to be made in the comments and not in the answers.
