A search for all primes where I limited the sum to 25000 digits revealed the following solutions:
I also searched for solutions where I reversed the order of the primes in base^exponent.
The smallest solution was the prime:
A table comprising all solutions where the sum has less than 10000 digits:
All numbers are strong pseudo primes, I used 11 Miller-Rabin tests. The primes with an (*) are certified prime.
Giorgios wrote:Starting with 2^3..., no other primes were found. (I checked up to 4000 consecutive primes)
If we start from an other prime than 2, here are some results (these sums contain only consecutive primes):
257^263 + ... + 431^433 is a 1141-digit prime (30 consecutive primes)
653^659 + ... + 1039^1049 is a 3165-digit prime (58 consecutive primes)
2333^2339 + ... + 3673^3677 is a 13109-digit prime (170 consecutive primes)
43^47 + ... + 1109^1117 is a 3402-digit prime (174 consecutive primes)
41^43 + ... + 1621^1627 is a 5223-digit prime (246 consecutive primes)
757^761 + ... + 4049^4051 is a 14614-digit prime (426 consecutive primes)
67^71 + ... + 4933^4937 is a 18233-digit prime (642 consecutive primes)
Q1: I didn't find a new prime or prp up to ...+15413^15427
Found no more solutions up to last term 37747^37781.