Phil,

Given that the mean nb of primes per doubling is now 0.19, the probability of finding a yet undiscovered PRP by double-checking the last sequence (40291) in the relevant range (1.25M to 5.2M) is quite low, around 6%*(1-exp(-0.19*LN(5.2/1.25)/LN(2))) = 2%. Thus for this last sequence double checking would better be considered in the future by managing the expected increasing error rate along with increasing exponent.

OTOH for the other sequences I would suggest to make a double checking up to the discovered PRP exponent in order to ensure that it is indeed the lowest exponent ("Keller prime"). I think it would not be an unbearable task, it could result in easing the search for prime proof, and if clearly mentioned, would spare dispersed efforts of finding such hypothetical lower PRPs. Don't forget that for k=67607 the exponent 46549 had been discovered before 16389.