Well, the probability of a randomly chosen [in some range] ODD integer being prime is going to be twice the probability for a randomly chosen integer [in that range]. So maybe you're really only trying to explain 5/4. --ms Dan Asimov wrote:
I (well, OK, a C program) factored Q(N) = N^2+N+1 for 10000 <= N < 11000, and found there to be 128 primes among these 1000 numbers.
Note: x/ln(x) at Q(10^4) is ~ 5429194, whereas x/ln(x) at Q(10^4) + 1000 ~ 5429245.
So the Prime Number theorem estimates about 51 primes for that run of 1000 consecutive numbers. This should be an upper bound, with high probability, for the # of primes among 1000 integers K selected randomly with Q(10^4) <= K < Q(10^4+10^3).
But the actual count is 128, about 5/2 that number.
Can the absence of prime factors of the form 6k-1 adequately explain such a high number of primes?
--Dan A.
-------------------------------------------------------------------- Don Reble djr@nk.ca writes:
I wrote:
... factorization of Q(N) = N^2+N+1 ... there appears to be an unusually high frequency of Q(N)'s being prime. And also when Q(N) is composite, it appears as though there tend to be unusually few factors (and unusually large ones).
Is there a theory that would confirm these suspicions?
I might know a little bit about that...
Q is the third cyclotomic polynomial, a factor of N^3-1, and Q(N) can't be divisible by a 6k-1 prime. Since there are fewer prime factors to choose from, you get fewer factors, and more primes. See also http://www.asahi-net.or.jp/~KC2H-MSM/cn
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