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Old 2021-08-26, 18:44   #8
Dobri
 
"Καλός"
May 2018

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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.
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