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

*π*_{3,2}(

*x*) and

*π*_{3,1}(

*x*) are equal.

Actually,

*π*_{3,2}(

*x*) =

*π*_{3,1}(

*x*) for

*x* = 2, 3, 7, 13, 19, 37, 43, 79, 163, 223 and 229:

*π*_{3,2}(2) =

*π*_{3,1}(2) = 0 (trivial case)

*π*_{3,2}(3) =

*π*_{3,1}(3) = 0 (trivial case)

*π*_{3,2}(7) =

*π*_{3,1}(7) = 1

*π*_{3,2}(13) =

*π*_{3,1}(13) = 2

*π*_{3,2}(19) =

*π*_{3,1}(19) = 3

*π*_{3,2}(37) =

*π*_{3,1}(37) = 5

*π*_{3,2}(43) =

*π*_{3,1}(43) = 6

*π*_{3,2}(79) =

*π*_{3,1}(79) = 10

*π*_{3,2}(163) =

*π*_{3,1}(163) = 18

*π*_{3,2}(223) =

*π*_{3,1}(223) = 23

*π*_{3,2}(229) =

*π*_{3,1}(229) = 24

It seems that the problem is still open for

*x* -> Infinity as there is no strict proof.