The original paper by Bays and Hudson (1978, available in JSTOR, see
https://www.jstor.org/stable/2006165...65734d170a779e) has no mention of several cases for small primes when the numbers of primes of the two types
π3,2(
x) and
π3,1(
x) are equal.
Actually,
π3,2(
x) =
π3,1(
x) for
x = 2, 3, 7, 13, 19, 37, 43, 79, 163, 223 and 229:
π3,2(2) =
π3,1(2) = 0 (trivial case)
π3,2(3) =
π3,1(3) = 0 (trivial case)
π3,2(7) =
π3,1(7) = 1
π3,2(13) =
π3,1(13) = 2
π3,2(19) =
π3,1(19) = 3
π3,2(37) =
π3,1(37) = 5
π3,2(43) =
π3,1(43) = 6
π3,2(79) =
π3,1(79) = 10
π3,2(163) =
π3,1(163) = 18
π3,2(223) =
π3,1(223) = 23
π3,2(229) =
π3,1(229) = 24
It seems that the problem is still open for
x -> Infinity as there is no strict proof.