Prime Numbers

1.6 Research problems 81
1.2.1. A reference is [Hensley and Richards 1973]. Remarkably, those authors
showed that on assumption of the prime k-tuples conjecture, there must exist
some y for which
π(y + 20000) − π(y) >π(20000).
What will establish incompatibility is a proof that the interval (0, 20000]
contains an “admissible” set with more than π(20000) elements. A set of
integers is admissible if for each prime p there is at least one residue class
modulo p that is not represented in the set. If a finite set S is admissible, the
prime k-tuples conjecture implies that there are infinitely many integers n such
that n + s is prime for each s ∈ S. So, the Hensley and Richards result follows
by showing that for each prime p ≤ 20000 there is a residue class ap such that
if all of the numbers congruent to ap modulo p are cast out of the interval
(0, 20000], the residual set (which is admissible) is large, larger than π(20000).
A better example is that in [Vehka 1979], who found an admissible set of 1412
elements in the interval (0, 11763], while on the other hand, π(11763) = 1409.
In his master’s thesis at Brigham Young University in 1996, N. Jarvis was
able to do this with the “20000” of the original Hensley-Richards calculation
cut down to “4930.” We still do not know the least integer y such that (0,y]
contains an admissible set with more than π(y) elements, but in [Gordon and
Rodemich 1998] it is shown that such a number y must be at least 1731.
For guidance in actual computations, there is some interesting analysis of
particular dense admissible sets in [Bressoud and Wagon 2000]. S. Wagon has
reduced the “4930” of Jarvis yet further, to “4893.” The modern record for
such bounds is that for first y occurrence, 2077