Hello, primenumbers group members. These are not questions about large primes.
A) I am seeking a power of 2 that has in decimal a prime number of copies of each digit (in its decimal representation). I would expect this to have already been asked. I have an early collection of 6 values with one exceptional digit, another 6 with two exceptional digits, and then it is not even empirically obvious there are not just a finite number with three exceptions (while, I believe, there should be infinitely many answers to the original question). New question, already solved, or in between?
B) I have 102 cases so far of primes that make the counts of primes in 3/4 of the residue classes other than 0 modulo 5 have equal totals. The last came very slowly in relative terms and has been followed by an even longer lag. There should be an infinity of these cases, right? And I should actually expect that whether I can find it or not there are also ones giving all 4, or no? This cannot possibly (?) be a new exploration. [Incidentally, does anybody have good info on how the numbers for questions of this kind compare with random walks, theoretically? Is there anything at all beyond refinements in expected total counts?]
Note: Still waiting to see if my putative wager would have been correct, and I was able to convince myself that my guess about a reason for the final-digit coincidence was reasonable (but did not go further than to see if it was totally off).
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