Hardy-Littlewood constants for quadratic polynomials

[mart_r]

The polynomial x²+x+a with integer x and odd integer a has a certain density of prime numbers which can be measured by an associated Hardy-Littlewood constant, HL(x²+x+a).
Example: HL(x²+x+1) = 2.241465507..., loosely speaking that x²+x+1 has a probability about 2.24 times as high to be prime than a random number of the same size.

Did I get that right that for every r that divides 4a-1 and every non-negative integer n, HL(x²+x+ (1+(4a-1)*r²ⁿ)/4) is equal?
What broader context can this be viewed in?

[mart_r]

Follow-up question:

Is there a similar measure for the numbers of the form 2kp+1 yielding a lot of composites, in which case the generalized repunits (b^p-1)/(b-1) would produce an above-average number of primes?
Successive records of this form should be* formed by the primes p = {3, 5, 7, 13, 17, 19, 31, 59, 109, 157, 167, 317, 457, 521, 1163, 1741, 1997, 2053, 3079, 3833, 5227, 5641, 11069, 12919, 13469, 14419, 16103, 19813, 19891, 22441, 22691, 28229, 30391, 31667, 37189, 39097, 39829, 51413, 71593, 74507, 85627, 93607, 104801, 117899, 138163, 170167, 216091, ...}
* not all numbers in the list may stand the test of a more sophisticated computation, but I'm confident that most of them will

[sweety439]

Originally Posted by mart_r

Follow-up question:

Is there a similar measure for the numbers of the form 2kp+1 yielding a lot of composites, in which case the generalized repunits (b^p-1)/(b-1) would produce an above-average number of primes?
Successive records of this form should be* formed by the primes p = {3, 5, 7, 13, 17, 19, 31, 59, 109, 157, 167, 317, 457, 521, 1163, 1741, 1997, 2053, 3079, 3833, 5227, 5641, 11069, 12919, 13469, 14419, 16103, 19813, 19891, 22441, 22691, 28229, 30391, 31667, 37189, 39097, 39829, 51413, 71593, 74507, 85627, 93607, 104801, 117899, 138163, 170167, 216091, ...}
* not all numbers in the list may stand the test of a more sophisticated computation, but I'm confident that most of them will

I think that you mean the records of https://oeis.org/A035095

However, (b^127-1)/(b-1) also produce a much-above-average number of primes, but 4*127+1 is prime, I really do not know why.

Indeed, (b^p-1)/(b-1) would produce an above-average number of primes for these primes, and a below-average number of primes for primes p such that 2*p+1 is prime (Sophie-Germain primes) (since (b^p-1)/(b-1) is divisible by 2*p+1 if b is a quadratic residue mod 2*p+1 and b != 1 mod 2*p+1), primes p such that 4*p+1 is prime, primes p such that 6*p+1 is prime

You can see https://oeis.org/A066180, which is the smallest base b such that the generalized repunit (b^p-1)/(b-1) is prime, it will be below-average number for the primes in https://oeis.org/A035095, and above-average number for the Sophie Germain primes, indeed, 2^521-1 is prime, and 521*n+1 is composite for all n<32, also, in both base 10 and base 12, the repunit R317 is prime, and 317*n+1 is composite for all n<30, it is also notable that there are many bases such that the repunits R17, R19, R31 is prime, but not for R59, since 59 is an irregular prime, also above-average number for the irregular primes (https://oeis.org/A000928), e.g. for p=37, the smallest irregular prime, https://oeis.org/A066180 is 61, which is much above-average, another case is p=491, which combined with the fact that it is a Sophie Germain prime and an irregular prime (it is also notable that it is the smallest Sophie Germain irregular prime whose corresponding safe prime is a regular prime), https://oeis.org/A066180 is 514, which is much above-average, note that the two primes p which Mersenne wrongly said 2^p-1 is prime, 67 and 257, are both irregular primes, and all primes p < 257 such that 2^p-1 is prime are regular primes.

[mart_r]

Many thanks for your reply!

I had devised a scheme some years ago to compare how prolific the generalized repunits are w.r.t. the values of p, by taking a sum

\(s_r:=\sum_{q=7,\,q\,prime}^x \sum_{r:\,every\,odd\,prime\,factor\,of\,q-1} \frac{r-1}q\)

and then looking at \(\lim_{x\to\infty} s_r-\log\log x\).

s₃ ~ log log x - 0.7137809590188862582393
s₅ ~ log log x - 0.8353377999490359325575
s₇ ~ log log x - 1.1224508552674277645804
s₁₁ ~ log log x - 0.9713749289063147467859
s₁₃ ~ log log x - 1.2082293777700878099127
s₁₇ ~ log log x - 1.3685790169746073220730
s₁₉ ~ log log x - 1.6194280070050371094275
s₂₃ ~ log log x - 1.0578200394879716206056
s₂₉ ~ log log x - 1.1685981976748223373826
s₃₁ ~ log log x - 1.7000384501264852200143
etc.

The lower the asymptotic value of s_p (or, the larger the value that's being subtracted from log log x), the more likely it is for (b^p-1)/(b-1) to yield a lot of primes.

I'm not entirely sure whether this approach is justified, I'd have to dig a little deeper into the theory and also do some number crunching.

Originally Posted by sweety439

However, (b^127-1)/(b-1) also produce a much-above-average number of primes, but 4*127+1 is prime, I really do not know why.

s₁₂₇ has only the second lowest asymptotic value of all s_p with p <= 127, so it wouldn't surprise me if the prime output is quite high. Note that, for instance, 2*k*127+1 is composite for 3 <= k <= 8.

[sweety439]

Originally Posted by mart_r

Many thanks for your reply!

I had devised a scheme some years ago to compare how prolific the generalized repunits are w.r.t. the values of p, by taking a sum

There is a generalization of the generalized repunit (b^p-1)/(b-1): the number Phi(n,b), where Phi is the cyclotomic polynomial, and since the degree of Phi(n,b) is eulerphi(n), the average number (the Hardy-Littlewood constant for cyclotomic polynomials) will be 1/eulerphi(n)

I indeed have a data for the 2<=b<=4096 such that Phi(n,b) is prime (ignore the numbers with "*", "**", or "***", and plus the prime Phi(6,2), I make this data is for the generalized unique primes), there will a below-average number of primes Phi(n,b) if n+1 is prime, or 2*n+1 is prime, or 3*n+1 is prime, or 4*n+1 is prime, and an above-average number of primes Phi(n,b) if k*n+1 is composite for all small k (see https://oeis.org/A034694 and https://oeis.org/A034693 and https://oeis.org/A120857), also, you can see the sequence https://oeis.org/A085398, a(n) will be large if n+1 is prime, or 2*n+1 is prime, or 3*n+1 is prime, or 4*n+1 is prime, and small (e.g. a(n) = 2 or 3) if k*n+1 is composite for all small k (a(n) is usually (but not always) not perfect power (I call the n such that a(n) is perfect power as "unusual numbers", such numbers n are 20, 28, 44, 66, 74, 87, 92, 96, 104, 138, 140, 148, 152, 156, 166, 178, 182, 189, 204, 210, 232, 249, 250, 264, 268, 292, 298, 300, ...), and if a(n) = m^r with r>1, then all prime factors of r divide n, and a(n*d) = m^(r/d) for all d dividing r), a(n) ~ gamma*eulerphi(n) (I conjectured), and I called the n such that a(n) > eulerphi(n) as "Satan numbers", such numbers n are 1, 2, 25, 37, 44, 68, 75, 82, 99, 115, 119, 125, 128, 159, 162, 179, 183, 188, 203, 213, 216, 229, 233, 243, 277, 289, 292, 301, 309, 316, 323, 329, 362, 376, 391, 397, 421, 423, 427, 432, 435, 438, 442, 443, 444, 459, 466, 473, 484, 491, 494, ...

[mart_r]

OK, thanks!
It's interesting to note that the prime terms in A120857 are similar to those I gave in post # 2. Maybe compute more terms of A120857 and see how many more coincide with my list?
Until then, I'm going to need some time to digest all about cyclotomic polynomials...


P.S.: HL(x²+x+a(n)) is the same for all terms a(n) in A066443. Just in case anyone thinks this is worthy to add in the comments.

[sweety439]

Does Hardy-Littlewood constant also exist for exponential sequences (a*b^n+c)/gcd(a+c,b-1) (with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1, and variable integer n>=1)?

[mart_r]

Maybe so. But those would depend on two or three different variables, I reckon that they would be much harder to calculate. I don't yet have the prerequisites to figure out efficient ways to do that, certainly others already did a lot of work on that, unfortunately I don't know where to start looking for it (at least for the special form of numbers you gave).