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\begin{center}
\vskip 1cm{\LARGE\bf Twin Prime Statistics}
\vskip 1cm
\large
Harvey Dubner \\
85 Long Hill Road \\
Oakland, NJ  07436 \\
USA\\
\href{mailto:harvey@dubner.com}{\tt harvey@dubner.com}\\
\end{center}

\vskip .2 in
\begin{abstract}
Hardy and Littlewood conjectured that the the number of twin primes
less than $x$ is asymptotic to 
$2C_2\int_{2}^{x}\frac{dt}{(\log t)^{2}}$
where $C_2$ is the twin prime constant.  This has been shown
to give excellent results for $x$ up to about $10^{16}$.  This article
presents statistics supporting the accuracy of the conjecture up to
$10^{600}$.
\end{abstract}

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\section {Introduction}
The Twin Prime Conjecture asserts that there are an infinite number of twin primes. 
Proving this conjecture remains one of the great unsolved problems in mathematics.  
In 1923, Hardy and Littlewood \cite{HL23} conjectured that the number of twin primes
less than $x$ is
\begin{equation}
L_{2}(x)\sim 2C_{2}\int_{2}^{x}\frac{dt}{(\log t)^{2}}\quad ,
\end{equation}
where
\begin{equation}
C_2=\prod_{p \text{ prime}>2}\left(1-\frac{1}{(p-1)^2}\right)= 0.6601618158\cdot\cdot\cdot
\end{equation}
is the ``twin-prime constant." The reasoning that leads to $C_2$ has been explained by 
many authors and is described by Ribenboim in \cite{PR04}.  
Although this conjecture (1.1) has never been proved, 
it gives remarkably good results.  For example, Crandall and Pomerance \cite[p13]{CP} 
report that by actual count { } 
\begin{equation*}
\pi_2(2.75\cdot 10^{15})=3049989272575,
\end{equation*}
while from eq.(1.1)
\begin{equation*}
L_2(2.75\cdot 10^{15})\approx 3049988592860.
\end{equation*}  
In this article we present statistics supporting the accuracy of conjecture (1.1)
up to $10^{600}$.

\section{Method}

When dealing with large numbers
it is obvious that using exact counts of all twin primes eventually becomes 
impractical, so that incremental counts have to be substituted for total counts. 
Equation (1.1) becomes
\begin{equation}
L_2(x_0,x_0+x)\sim 2C_{2}\int_{x_0}^{x_0+x}\frac{dt}{(\log t)^{2}}
\approx 2C_2\frac{x}{(\log x_0)^{2}}   \quad , 
\end{equation}
which assumes that $x_0$ is large enough so that $\log x$ is virtually constant 
over $x$.  

To find the number of twin primes in $x$ we actually count all the prime gaps in $x$.
This is accomplished by processing segments of 256,000,000 numbers, first by sieving 
to eliminate all composite numbers with a prime factor less than $10^9$, then by
subjecting the remaining numbers to a Fermat test for primality. Those that 
pass the Fermat test are assumed to be prime even though there is a slight chance of 
a small number being composite. However, any such error would not have a significant
effect on the twin prime count. See below for a discussion of this topic.

We started collecting gap counts at $x_0=10^{20}$.  Finding $10^9$ gaps, of 
which 28,667,923 were twins, took about 2 hours on a 2.0 GHz computer. This required
the determination of the primality of about $23\cdot10^9$ odd numbers.  The same 
gap count problem starting at $x_0=10^{50}$ took about 82 hours.  

\subsection{Pseudoprimes}
A number which passes a base $b$ Fermat test might not be prime but could be  
psp, a base $b$ pseudoprime.  There have been various studies estimating psp  
frequencies but the results are asymptotic in nature and usually do not give 
useful information for relatively small numbers. We investigated psp's near $10^{19}$
because we could use 64-bit arithmetic to sieve such numbers so as to accurately and 
efficiently determine the composite numbers.  These were all subjected to a 
base 3 Fermat test to determine which were psp's.

$2000\cdot10^9$ consecutive odd numbers near $10^{19}$ were tested and 17 psp's were found.  Thus 
the probability of an odd number near $10^{19}$ being a psp is about
\begin{equation*}
\text{Prob(psp)} = 17/(2000\cdot10^9) = 8.5\cdot 10^{-12}\quad.
\end{equation*}
Applying the psp probability for $10^{19}$ to the $23\cdot10^9$ odd numbers above 
$10^{20}$ gives an expected error of 
\begin{equation*}
\text{expected error}=(23\cdot10^9)\cdot(8.5\cdot10^{-12})=0.20
\end{equation*}
and so we would expect to have at most one extra twin prime gap appearance
in error when counting $10^9$ gaps.  Actually 
the error is considerably less than that shown since the sieving process would 
have eliminated most of
the psp's.  For larger values of $10^n$ the number of expected psp's 
continues to decrease,
therefore it is clear that using a Fermat test to determine twin prime
statistics is well justified.  

Insisting on absolute accuracy in determining 
primes would have increased the computer time by a factor of 
at least 1000, making this project virtually impossible.   
      
\subsection{Poisson distribution}
Although the exact distribution of counts of twin primes is not known, it seems 
reasonable to assume that such counts might be approximated by a Poisson probability 
distribution since this is true for almost all distributions of rare phenomena.  
We could then present the error between the actual and estimated twin prime counts 
as the number of standard deviations which effectively normalizes the error.
For the Poisson distribution the standard deviation is the square root of the 
twin prime count.
Using a common probability distribution to describe twin prime counts is not new.  
For example, R. Brent used that approach in a 1975 paper \cite{RB75}, as did 
T.Nicely in 1999 in evaluating Brun's constant \cite{TN99}. We are only using 
probability concepts to normalize the error in predicting twin prime counts. 

\section {results}
The twin prime counts shown here were obtained as part of a larger project to obtain 
the frequencies of all prime gaps near $10^n$ for a wide range of $n$. For 
a given $n$, gap data was collected until the total of all gaps, $g_t$, equalled a 
selected limit such as $10^9$. Because of the Prime Number Theorem this means 
that about $g_t\cdot\log(10^n)$ numbers have to be checked for primality. Substituting 
this into eq.(2.1),
\begin{equation}
L_2(10^n,10^n+g_t\cdot\log(10^n))\approx 2C_2\frac{g_t\cdot\log(10^n)}{(\log(10^n))^2}
=2C_2\frac{g_t}{\log(10^n)}
\end{equation}
which is the predicted number of twin primes, shown in column four of Tables 1 and 2.   
The error measured in Standard Deviations,
\begin{equation*}
SD=\frac{\text{error}}{\sqrt{\text{actual}}}
\end{equation*}
is shown in the last column. 

It is clear from examining the data in Tables 1 and 2 that the Hardy Littlewood 
conjecture for the count of twin primes, eq. (1.1), is ``accurate" up to primes
of 600 digits.  The error is consistantly of the order of the square root of the
actual twin prime count.  Unfortunately we do not yet have any theory for improving 
the error prediction.   

     
  

\begin{center}
\begin{table}[htb]
\centering
\caption{Twin Prime data for $10^{20}$ to $10^{55}$ }
\label{tab:1}
\vskip .15in
\begin{tabular}{rrrrrrr}
%\multicolumn {7} c {total number of gaps per column = 1,000,000,000}\\\\

 & total& \multicolumn {2} c {twin primes}& & percent& error \\
size& gaps& actual& predicted& error& error& in SD\\
\hline
$10^{ 20 }$&$10^9$& 28667923 & 28670463 &-2540 &  -0.0089&  -0.47\\
$10^{ 21 }$&$10^9$& 27297642 & 27305203 &-7561 &  -0.0277&  -1.45\\
$10^{ 22 }$&$10^9$& 26069405 & 26064057 & 5348 &   0.0205&   1.05\\
$10^{ 23 }$&$10^9$& 24933403 & 24930837 & 2566 &   0.0103&   0.51\\
$10^{ 24 }$&$10^9$& 23885814 & 23892052 &-6238 &  -0.0261&  -1.28\\
$10^{ 25 }$&$10^9$& 22942786 & 22936370 & 6416 &   0.0280&   1.34\\
$10^{ 26 }$&$10^9$& 22060593 & 22054202 & 6391 &   0.0290&   1.36\\
$10^{ 27 }$&$10^9$& 21243727 & 21237380 & 6347 &   0.0299&   1.38\\
$10^{ 28 }$&$10^9$& 20482302 & 20478902 & 3400 &   0.0166&   0.75\\
$10^{ 29 }$&$10^9$& 19773012 & 19772733 & 279 &   0.0014&   0.06\\
$10^{ 30 }$&$10^9$& 19113536 & 19113642 &-106 &  -0.0006&  -0.02\\
$10^{ 31 }$&$10^9$& 18503641 & 18497073 & 6568 &   0.0355&   1.53\\
$10^{ 32 }$&$10^9$& 17913954 & 17919039 &-5085 &  -0.0284&  -1.20\\
$10^{ 33 }$&$10^9$& 17372782 & 17376038 &-3256 &  -0.0187&  -0.78\\
$10^{ 34 }$&$10^9$& 16868189 & 16864978 & 3211 &   0.0190&   0.78\\
$10^{ 35 }$&$10^9$& 16387161 & 16383121 & 4040 &   0.0247&   1.00\\
$10^{ 36 }$&$10^9$& 15931851 & 15928035 & 3816 &   0.0240&   0.96\\
$10^{ 37 }$&$10^9$& 15496561 & 15497547 &-986 &  -0.0064&  -0.25\\
$10^{ 38 }$&$10^9$& 15090215 & 15089717 & 498 &   0.0033&   0.13\\
$10^{ 39 }$&$10^9$& 14707109 & 14702801 & 4308 &   0.0293&   1.12\\
$10^{ 40 }$&$10^9$& 14334329 & 14335231 &-902 &  -0.0063&  -0.24\\
$10^{ 41 }$&$10^9$& 13982239 & 13985591 &-3352 &  -0.0240&  -0.90\\
$10^{ 42 }$&$10^9$& 13650736 & 13652601 &-1865 &  -0.0137&  -0.50\\
$10^{ 43 }$&$10^9$& 13337659 & 13335099 & 2560 &   0.0192&   0.70\\
$10^{ 44 }$&$10^9$& 13029542 & 13032028 &-2486 &  -0.0191&  -0.69\\
$10^{ 45 }$&$10^9$& 12740605 & 12742428 &-1823 &  -0.0143&  -0.51\\
$10^{ 46 }$&$10^9$& 12471625 & 12465418 & 6207 &   0.0498&   1.76\\
$10^{ 47 }$&$10^9$& 12201271 & 12200197 & 1074 &   0.0088&   0.31\\
$10^{ 48 }$&$10^9$& 11948039 & 11946026 & 2013 &   0.0168&   0.58\\
$10^{ 49 }$&$10^9$& 11703340 & 11702229 & 1111 &   0.0095&   0.32\\
$10^{ 50 }$&$10^9$& 11464673 & 11468185 &-3512 &  -0.0306&  -1.04\\
$10^{ 55 }$&$10^9$& 10426568 & 10425623 & 945 &   0.0091&   0.29\\
\end{tabular}
%note: Standard deviation near $10^{35} = 8000$ (approx)
\end{table} 
\end{center}

\begin{center}
\begin{table}[htb]
\centering
\caption{Twin Prime data for $10^{60}$ to $10^{600}$ }
\vskip .15in
\label{tab:2}
\begin{tabular}{rrrrrrr}
%\multicolumn {7} c {total number of gaps per column = 1,000,000,000}\\\\

 & total& \multicolumn {2} c {twin primes}& & percent& error \\
size& gaps& actual& predicted& error& error& in SD\\
\hline
$10^{ 60 }$&$10^8$& 956273 & 955682 & 591 &   0.0618&   0.60\\
$10^{ 70 }$&$10^8$& 819450 & 819156 & 294 &   0.0359&   0.32\\
$10^{ 80 }$&$10^8$& 716327 & 716761 &-434 &  -0.0606&  -0.51\\
$10^{ 90 }$&$10^8$& 636623 & 637121 &-498 &  -0.0782&  -0.62\\
$10^{ 100 }$&$10^8$& 572885 & 573409 &-524 &  -0.0915&  -0.69\\
$10^{ 110 }$&$10^8$& 520799 & 521281 &-482 &  -0.0926&  -0.67\\
$10^{ 120 }$&$10^8$& 477439 & 477841 &-402 &  -0.0842&  -0.58\\
$10^{ 130 }$&$10^8$& 440294 & 441084 &-790 &  -0.1794&  -1.19\\
$10^{ 140 }$&$10^8$& 409384 & 409578 &-194 &  -0.0474&  -0.30\\
$10^{ 150 }$&$10^8$& 382170 & 382272 &-102 &  -0.0267&  -0.16\\
$10^{ 160 }$&$10^7$& 36006 & 35838 & 168 &   0.4666&   0.89\\
$10^{ 180 }$&$10^7$& 32107 & 31856 & 251 &   0.7818&   1.40\\
$10^{ 200 }$&$10^7$& 28652 & 28670 &-18 &  -0.0628&  -0.11\\
$10^{ 220 }$&$10^7$& 26213 & 26064 & 149 &   0.5684&   0.92\\
$10^{ 240 }$&$10^7$& 23638 & 23892 &-254 &  -1.0745&  -1.65\\
$10^{ 260 }$&$10^7$& 22281 & 22054 & 227 &   1.0188&   1.52\\
$10^{ 280 }$&$10^7$& 20458 & 20478 &-20 &  -0.0978&  -0.14\\
$10^{ 300 }$&$10^7$& 19181 & 19113 & 68 &   0.3545&   0.49\\
$10^{ 320 }$&$5\cdot10^6$& 9041 & 8959 & 82 &   0.9070&   0.86\\
$10^{ 340 }$&$5\cdot10^6$& 8584 & 8432 & 152 &   1.7707&   1.64\\
$10^{ 360 }$&$5\cdot10^6$& 8090 & 7964 & 126 &   1.5575&   1.40\\
$10^{ 380 }$&$5\cdot10^6$& 7410 & 7544 &-134 &  -1.8084&  -1.56\\
$10^{ 400 }$&$5\cdot10^6$& 6997 & 7167 &-170 &  -2.4296&  -2.03\\
$10^{ 450 }$&$11\cdot10^6$& 14035 & 14016 & 19 &   0.1354&   0.16\\
$10^{ 500 }$&$7\cdot10^6$& 8140 & 8027 & 113 &   1.3882&   1.25\\
$10^{ 550 }$&$5\cdot10^6$& 5084 & 5212 &-128 &  -2.5177&  -1.80\\
$10^{ 600 }$&$5\cdot10^6$& 4734 & 4778 &-44 &  -0.9294&  -0.64\\
\end{tabular}
\end{table} 
\end{center}

\section{Acknowledgements}
We wish to thank Jens Kruse Andersen for the use of his excellent program for counting 
pseudoprimes near $10^{19}$.

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\begin{thebibliography}{1}

\bibitem{RB75}
R.~P.~Brent, Irregularities in the distribution of primes and twin primes,
\emph {Math. Comp.} \textbf{29}, (1975), 43--56.

\bibitem{CP}
R.~Crandall and C.~Pomerance, \emph{Prime Numbers: a Computational Perspective},
Springer-Verlag, New York, 2001.

\bibitem{HL23}
G.~H.~Hardy and J.~E.~Littlewood, Some problems on partitio numerorum, III: 
On the expression of a number as a sum of primes.
\emph{Acta Math.} \textbf{44}, (1923), 1--70.

\bibitem{TN99}
T.~Nicely, Enumeration to $1.6*10^{15}$ of the twin primes and Brun's constant, (1999).
\textbf{http://www.trnicely.net/twins/twins2.html} 


\bibitem{PR04}
 P.~Ribenboim, \emph{The Little Book of Bigger Primes}, second edition,
Springer-Verlag, New York, 2004.

\end{thebibliography}



\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B05; Secondary 11A41.

\noindent \emph{Keywords:}
prime gaps, twin primes, twin prime constant.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A001097},
\seqnum{A001359}, and
\seqnum{A006512}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received May 3 2005;
revised version received  August 25 2005.
Published in {\it Journal of Integer Sequences}, August 29 2005.

\bigskip
\hrule
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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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