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\line{Paul Coddington,\hfill}
\line{Physics Department,\hfill}
\line{Syracuse University,\hfill}
\line{Syracuse, NY 13244.\hfill}
\line{10/6/90\hfill}
\bigskip
\bigskip
\noindent
Dear Referee,
\bigskip

I would like to make a couple of comments concerning your objections
to my paper``A Computer Study of the Goldbach Curve''. 
Upon re-examining my program in light of your comments, I found that you are
correct that the program I used was very inefficient, and I estimate that I 
could have obtained the same results with a factor of about 7 less run time.
Alternatively I could
have extended the region of study up to about $5 \times 10^7$ rather than
$2 \times 10^7$, which would not have greatly altered the results.
Each of the processors in the machines used was about 3 times less powerful 
than a Sun3, so it should have taken me approximately 90 hours on 20 
processors, which is roughly equivalent to 600 hours on a Sun3, and thus
agrees with your figures. You are certainly correct that there was no
great computational or programming artistry involved.

\bigskip
I would, however, disagree that the results do not represent any increase in
understanding.
While it is certainly true that there is no doubt as to the validity of the
Goldbach Conjecture, that is not the main problem which the paper is
addressing. Although the Hardy-Littlewood asymptotic is also almost certainly
true, I would disagree that it is unworthy of a stringent test, since
the only previous test (by Shah and Wilson, as mentioned in the paper of
Littlewood and Hardy)
was very cursory, covering only 35 values of $n$.
Also, virtually nothing was known about how (or even whether) the error
decreases as a function of $n$,
and in fact Littlewood and Hardy state that one of the reasons
they failed to prove their conjecture was that they could not bring down the
bound on the error far enough. In the paper I show how the error decreases,
and even find a simple expression for
the error as a function of $n$, which I would claim is an increase
in understanding of the Hardy-Littlewood conjecture.

\bigskip
I also propose and check a lower bound for the Goldbach curve, which to my
knowledge has not been done before. Another new result is that the number of
solutions $n$ of $N_2(n) = k$ can be bounded below by an increasing function
of $k$ in the region studied, which provides strong new quantitative evidence
for the truth of the conjecture that the above equation has at least one
solution for all $k > 0$. Perhaps I should have emphasized these points more
in the paper.

\bigskip
Yours sincerely,

\vfill
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