Asymptotics for primality of sum of three consecutive primes - MathOverflow

Asymptotics for primality of sum of three consecutive primes

2011-08-10T01:50:42Z

We consider the sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where ${p_i}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc..

The first few values of $R_n$ are: 10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 425, 439, 457, ...

Now, we define $R(n)$ to be the number of prime numbers in the set ${R_0, R_1 , \dots , R_n}$.

What I have found (without justification) is that $R(n) \approx \frac{2n}{\ln (n)}$.

My lack of programming skills, however, prevents me from checking further numerical examples. I was wondering if anyone here had any ideas as to how to prove this assertion.

As a parting statement, I bring up a quote from Gauss, which I feel describes many conjectures regarding prime numbers: "I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of."

Answer by GH for Asymptotics for primality of sum of three consecutive primes

2011-08-10T02:13:11Z

I believe proving (or disproving) such a statement is beyond current technology. On the other hand, by a crude heuristics the conjecture must be right: $R_1,\dots,R_n$ are $n$ odd numbers up to $\sim 3n\ln(n)$ which are rather evenly distributed in size and in residue classes. In this range the density of primes among odd numbers is $\sim 2/\ln(n)$, so $R(n)$ should be $\sim 2n/\ln(n)$.

EDIT: As Noam Elkies points out in a comment below, there should be a fine tuning similar to the twin prime constant.

Answer by Álvaro Lozano-Robledo for Asymptotics for primality of sum of three consecutive primes

2011-08-10T04:21:35Z

Answer by joriki for Asymptotics for primality of sum of three consecutive primes

2011-08-10T06:17:52Z

I verified Noam's calculation of the factor $\lambda=2.30096\ldots$ and Álvaro's computations, extended the latter and calculated the corresponding ratios:

$$ \begin{array}{|c|c|c|c|c|} n & R(n) & 2n/\log n & \lambda n / \log n & R(n)\log n/n\\ \hline 10 & 7 & 9 & 10 & 1.61181\\ 100 & 44 & 43 & 50 & 2.02627\\ 1000 & 339 & 290 & 333 & 2.34173\\ 10000 & 2437 & 2171 & 2498 & 2.24456\\ 100000 & 18892 & 17372 & 19986 & 2.17502\\ 1000000 & 157183 & 144765 & 166549 & 2.17156\\ 10000000 & 1346797 & 1240841 & 1427564 & 2.17078\\ 30000000 & 3784831 & 3484987 & 4009410 & 2.17208\\ \end{array} $$

(The values in the third and fourth columns are rounded to the nearest integer, the values in the last column are rounded to 5 digits after the decimal point.)

I don't think we can deduce anything from the ratio in this form, however, since it shows convergence in the "random" fluctuations but not with respect to the asymptotic approximations made, e.g. dropping a term $\log\log n$, which at this stage is still comparable to $\log n$; a more detailed analysis will be required to test the independence hypothesis in this case.

[Update:] With reference to Noam's comments below, here are some data for the relative frequencies of the sum of three consecutive primes being divisible by the first four odd primes. These are averaged over samples of $400,000$ primes beginning at powers of ten, which are given in the first column; note that these refer to the numbers $x$ themselves, not the indices $n$ of the primes.

$$ \begin{array}{|c|c|c|c|c|} \log_{10}x&3&5&7&11\\ \hline 8 &0.183&0.165&0.130&0.087\\ 9 &0.189&0.169&0.131&0.087\\ 10&0.195&0.170&0.133&0.087\\ 11&0.198&0.172&0.133&0.088\\ 12&0.203&0.173&0.133&0.088\\ 13&0.208&0.175&0.134&0.087\\ \hline \text{limit?}&0.250&0.188&0.139&0.090 \end{array} $$

I also looked at the joint distribution of the residues modulo $3$ for the three primes. There's a significant preference for avoiding repeated residues; for instance, at $x=10^9$, the repeating patterns $1,1,1$ and $2,2,2$ have relative frequencies around $0.095$, the alternating patterns $1,2,1$ and $2,1,2$ have relative frequencies around $0.150$, and the remaining mixed patterns have relative frequencies around $0.128$, which is almost completely explained by $1,1$ and $2,2$ having relative frequencies $0.445$ and $1,2$ and $2,1$ having relative frequencies $0.555$. I'm trying to work out a probabilistic model for these effects.