Prime Numbers

1.6 Research problems 75
1.6 Research problems
1.77. In regard to the Mills theorem (the first part of Theorem 1.2.2),
try
3j to find an explicit number θ and a large number n such that θ is prime
for j =1, 2,...,n. For example if one takes the specific rational θ = 165/92,
show that each of
θ 31
, θ 32
, θ 33
, θ 34
is prime, yet the number θ35 is, alas, composite. Can you find a simple
rational θ that has all cases up through n = 5 prime, or even further? Say a
(finite or infinite) sequence of primes q1 1 such that the sequence (⌊θ n ⌋) consists
entirely of primes? The existence of such a θ seems unlikely, yet the authors
are unaware of results along these lines. For θ = 1287/545, the integer
parts of the first 8 powers are 2, 5, 13, 31, 73, 173, 409, 967, each of which is
prime. Find a longer chain. If an infinite chain were to exist, there would
be infinitely many triples of primes p, q, r for which there is some α with
p = ⌊α⌋ ,q = α 2 ,r = α 3 . Probably there are infinitely many such triples
of primes p, q, r, and maybe this is not so hard to prove, but again the authors
are unaware of such a result. It is known that there are infinitely many pairs
of primes p, q of the form p = ⌊α⌋ ,q = α 2 ; this result is in [Balog 1989].
1.79. For a sequence A =(an), let D(A) be the sequence (|an+1 − an|). For
P the sequence of primes, consider D(P), D(D(P)), etc. Is it true that each of
these sequences begins with the number 1? This has been verified by Odlyzko
for the first 3 · 10 11 sequences [Ribenboim 1996], but has never been proved
in general.
1.80. Find large primes of the form (2 n +1)/3, invoking possible theorems on
allowed small factors, and so on. Three recent examples, due to R. McIntosh,
are
p =(2 42737 +1)/3, q =(2 83339 +1)/3, r =(2 95369 +1)/3.