Definition and Properties of Babylonian Primes
By means of his
interesting article
,
Vladislav Sinitsa from Russia
has drawn my attention to the so-called
Babylonian series of prime numbers
(Vavilonskij ryad prostykh chisel)
P(n) = 6*floor(n/2) + (-1)^{n+1},
for positive integers n >= 1.
Here, floor(x) denotes the biggest integer
smaller than or equal to a given real number x.
The Babylonian primes, briefly BP,
form a strictly increasing sequence
P(1) < P(2) < P(3) < ... < P(n) < ...
Except for the unit
P(1) = 1, which is not a prime,
the initial members of the sequence
P(2) = 5,
P(3) = 7,
P(4) = 11,
P(5) = 13,
P(6) = 17,
P(7) = 19,
P(8) = 23
are actually prime numbers.
However,
P(9) = 25 = 5*5
is the smallest example of a composite BP,
and, up to 100, there occur further composite BPs:
P(12) = 35 = 5*7,
P(17) = 49 = 7*7,
P(19) = 55 = 5*11,
P(22) = 65 = 5*13,
P(26) = 77 = 7*11,
P(29) = 85 = 5*17,
P(31) = 91 = 7*13,
P(32) = 95 = 5*19.
So, at the beginning, composite BPs are rather rare.
However, in higher ranges they occur quite frequently,
e. g.
P(39) = 115 = 5*23,
P(40) = 119 = 7*17,
P(41) = 121 = 11*11,
P(42) = 125 = 5*5*5
are four immediately consecutive composite BPs.
Due to the prime number theorem,
this behavior is to be expected:
for any upper bound B >= 1
the number of prime numbers p <= B
is approximately given by B/log(B),
whereas the number of BPs P(n) <= B is about B/3,
and (B/3) >> (B/log(B)), for sufficiently large B.
To enable a systematic search for composite BPs,
Vladislav Sinitsa suggests the following rule,
without giving a strict proof.
General Exclusion Rule:
If P(n) = a*b is a composite BP
with proper divisors 1 < a,b < a*b, for some n >= 1,
then all the BPs of the shape
P(n + 2*a*i) and P(n + 2*b*i), with i >= 1,
are also composite.
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