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Let $$S(t)=\sum_{k=1}^t p_k^{p_k}$$

Where $p_k$ is the kth prime number. E.g. $S(2)=2^2+3^3=31$

Since $S(2)$ and $S(4)$ were both the $p+2$ of a twin prime.

Continued search with Pari/GP to test to $t=10^4$ and have found primes only for $\{2,4,24\}$, $S(24)$ is not a twin.

Will $S(t)$ for $t>24$ always have a factor?

Asbjørn
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  • $S(10^4)$ has more than $500\ 000$ digits. This is a huge search limit and it will be difficult to find further primes. Since I guess that we cannot rule out small prime factors, a proof that there is no more such prime is probably out of reach. – Peter Feb 13 '21 at 17:21
  • I asked this question nearly $3$ years ago here – Peter Feb 14 '21 at 08:23
  • Thank you Peter, for the information.
    Is there an analytical way to approach this problem?     – Asbjørn Feb 16 '21 at 10:31
  • No, we can do nothing more than search for another prime (after applying trial division) – Peter Feb 16 '21 at 10:48

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