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Denote $$S(p):=2^2+3^3+5^5+\cdots +p^p$$

$S(p)$ is prime for $p=3,7,89$.

Is there another prime $p$ such that $S(p)$ is prime ? Is the number of primes $p$ such that $S(p)$ is prime, finite ?

Peter
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    I'm astonished there is a third case. I'm sure the probabilities indicate that the expected number is finite; effectively you have a number with no prior known divisibility roughly the size of $p^p$, which has a descending chance of being prime, $<(1/(p\ln p))$, in a population becoming sparse. – Joffan Feb 27 '18 at 01:02
  • Up to what $p$ did you check? – Tito Piezas III Feb 27 '18 at 02:20
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    Nothing more up to p < 10000 from what I see. – DanaJ Feb 27 '18 at 04:05
  • @DanaJ: For $p=10000$, then $S(p)$ has more than 40000 decimal digits. What primality test did you use? ECM? – Tito Piezas III Feb 27 '18 at 07:31
  • @TitoPiezasIII I arrived at $5500$ , but apparently, DanaJ got much further. – Peter Feb 27 '18 at 09:26
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    @TitoPiezasIII ECM is usually used to find factors. But with elliptic curves, prime proving is also possible. But the usual approach to decide whether a number is prime is the strong-probable-prime test. Only if a number passes it, we cannot be sure whether it is prime, although it is very likely in general. If a number fails that test, it must be composite. – Peter Feb 27 '18 at 12:05
  • @TitoPiezasIII as Peter said. I used ES BPSW, about 30 minutes total. There are simple pretests, a M-R base 2 test, then ES Lucas. Composites of this size will rarely make it past the M-R test. If one had passed then 40k digits would be more problematic for further testing, but it's easy for composites. – DanaJ Feb 27 '18 at 16:30
  • @DanaJ Hi, Dana! Do you have interest/time for EnzoCreti's prime-finding project ? Or can you tell me how I can concatenate two Mersenne-numbers in factordb ? (The later would be helpful as well) – Peter Feb 27 '18 at 16:34
  • Does anyone know the syntax for summation in OPEN PFGW ? – Peter Feb 27 '18 at 22:17
  • @Peter I don't know which one you had in mind -- Enzo has one or more questions every single day so it would be impossible to keep up. It's highly unlikely regardless as I'm already involved in prime gaps as well as generic software. I don't know the syntax for concatenation on factordb if it does exist. I hope you get an answer on your question about it. – DanaJ Feb 28 '18 at 07:43
  • @DanaJ I mean this : https://math.stackexchange.com/questions/2635516/numbers-n-of-the-form-10m2k%e2%88%9212k-1%e2%88%921-where-m-is-the-number-of/2636195#2636195 – Peter Feb 28 '18 at 08:31
  • I double-checked the range upto $p=9\ 973$ and found no further prime either. – Peter Feb 28 '18 at 11:18
  • @Peter I started the search but I think that it will take 2 months for finishing the job. – Enzo Creti Feb 28 '18 at 17:49
  • @EnzoCreti Did you also download PARI/GP ? – Peter Feb 28 '18 at 18:08
  • @Peter no but isnt it even slower than PFGW? – Enzo Creti Feb 28 '18 at 18:16
  • @EnzoCreti Yes, but you can also verify sums, products etc with is apparently not supported by PFGW or factordb. – Peter Feb 28 '18 at 19:54
  • @EnzoCreti What do you think abouth this : https://math.stackexchange.com/questions/2480379/smallest-prime-factor-of-lfloor-e-uparrow-e-uparrow-e-uparrow-e-rfloor ? – Peter Feb 28 '18 at 20:24
  • @Peter impressive! That number is huge! – Enzo Creti Feb 28 '18 at 20:30
  • Let us continue this discussion in chat. – Enzo Creti Mar 01 '18 at 12:48

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