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Old 2020-09-13, 13:39   #3
Dr Sardonicus
 
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Feb 2017
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I'd been meaning to look at this for a while. I finally got around to it.

In the present situation, we want p and q = 2*k*p + 1 both to be prime. The Bateman-Horn Conjecture says there is a constant C = C(k) such that the number of such p up to X is asymptotically

C*x/log2(X)

We also require q == 1 or 7 (mod 8). This introduces a "fudge" factor, giving

C*x/log2(X), for k == 0 (mod 4),

(1/2)*C*x/log2(X), for k == 1 (mod 4),

0, for k == 2 (mod 4), and

(1/2)*C*x/log2(X), for k == 3 (mod 4).

The restriction q == 1 or 7 (mod 8) means that 2 is a quadratic residue (mod q). Also, since k|(q-1), the finite field Fq contains q q-th roots of unity. We would expect that, among primes q having 2 as a quadratic residue, and q == 1 (mod k), 2 would be a 2k-th power residue of 1/k of them.

If we assume this proportion holds under the restriction that (q-1)/(2*k) is prime, we obtain a conjectural formula for the number of q up to X for which q = 2*k*p + 1 is a prime which divides Mp. Obviously, the factor 1/k reduces the constant multiplier if k > 1.

The formula for C is a bit complicated to go into here, but there is one aspect that may pertain to the value k = 12 being "favored."

If l is a prime not dividing 2*k, about 1/(l-1) of the primes p will be in the residue class -(2*k)-1 (mod l); l divides q = 2*k*p + 1 for these p. Thus, if 3 does not divide k, we can expect about half the values of 2*k*p + 1 to be divisible by 3. If 3 divides k, however, this possibility is eliminated.

The value k = 12 is thus thrice blessed -- it is divisible by 4, making the "fudge factor" as large as possible; the factor 1/k is not terribly small; and, the factor 3 eliminates the possibility of q being divisible by 3, which helps make the constant C bigger than it otherwise would be.

Someone more ambitious than I am might like to compute values of C for some given k's, and perhaps check how well the assumed factor 1/k fits the data.

Last fiddled with by Dr Sardonicus on 2020-09-13 at 13:41 Reason: grammar errors
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