@Carl Love

Firstly I think this subject has not been much studied since if it had there would be a trace in the oeis, and there isn’t.

Two sequences come to mind. The first is what you call PP(p), the number of prime pair inverses in Zp. The first few terms are: 0,0,1,1,0,1,1,2,1,0,1,2,2,2,2,2,2,3,4,3,2,3

and so on.

The second is the sequence a(n) = smallest prime p to have n prime inverse paires in Zp. The initial terms (if I’m not mistaken, are: 2,5,19,61,67,113,191...

neither of these appears in oeis so that to me says this is an open subject. Before my away trip I entered the second one as a draft in oeis but have not been able to finish it because of WiFi issues. I have no space to enter the PP(p) sequence because there is a big backlog, but I will do that as soon as a space comes up. Meantime the main intuitions have been yours since you have made a code and studied some of the data. Personally I was not entirely convinced by the independence idea, though somehow it did seem to have a plausible feel to it. Therefore I was ready to accept it as a working proposition, and in the absence of any other better idea to replace it with. Your intuition, achieved in a few moments of thought seems to have been quite a jump ahead and I for one have not yet seen how you proceeded from there to complete the proof. I also did not anticipate the Li function, my first thought in that direction would have been the prime counting function pi, which I know to be similar to Li. In any case your data and the bounds are quite convincing, so maybe you are right, but I agree the proof is looking tricky, and might only appear possible if, as you already suggest it can be shown to be true asymptotically, as p >>>oo.

Some other aspects : For p=29, Zp contains no prime inverse pairs and appears to be the largest prime with this property (2,3,11) also have it. I have only checked (manual Maple) out to p=191 but how to be sure there is not some weird prime way out there which has the same property? As p gets bigger the likelihood of this gets smaller and smaller of course, but it’s still a valid question. Interesting finite sequences are rare, and this might be one!

Similarly, for p=5, we find that every prime in Zp has a prime inverse. Again as p grows, the likelihood that another p crops up with this property reduces. And this is born out by the limited data I have produced. But being certain about it is another matter.

I will try and use your code tomorrow if time permits, and see what the extended data looks like. From that I will be able to extend the second sequence I mentioned.

Thanks again for your input here, I have found your intuitions, backed with data to be very credible. Maybe you (or I) could address such questions to the Seqfan group associated with the Oeis? I think they would be interested in this subject. In any case when these two sequences are approved, your input to edit them by addition of your findings would I suggest be a reasonable way to go. Ok I stop there, it’s been a long haul today and I have to sleep now.

regards

David