 # The Babylonian Series of Primes

## Are all of its members prime numbers ?

 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.

 There arise two questions about Babylonian primes: Open Problems: 1. Does every prime number, except 2 and 3, occur among the Babylonian primes P(n), for some n >= 1 ? 2. Does there exist a rigorous proof of the General Exclusion Rule ? I am looking forward to suggestions or solutions !

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