On the subsequence of primes having prime subscripts
نویسندگان
چکیده
There are a number of subsets of primes with a conjectured density of a constant times x/ log x. These include the primes separated by a fixed even integer, Sophie Germain primes and the so-called “thin primes”, i.e. primes of the form 2q − 1 where q is prime [3]. In order to gain some familiarity with sequences of this density we undertook an investigation of the set of primes of prime order, called here “prime-primes”, and report on the results of this investigation here. Some of the properties of this sequence are reasonably straight forward and the derivation follows that of the corresponding property for the primes themselves. Others appear to be quite deep and difficult. Of course the process of taking a prime indexed subsequence can be iterated, leading to sequences of primes of density x/ log x, x/ log x etc, but these sequences are not considered here. In Section 2 upper and lower bounds for the n prime-prime are derived, in Section 3 the prime-prime number theorem is proved with error bounds equivalent to that of the prime number theorem, in Section 4 an initial study of gaps between prime-primes is begun, and in Section 5 it is shown
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