%goldbach.tex:A  Proof of the Celebrated Goldbach's Theorem
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\bf
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{
A Proof of the Celebrated Goldbach's Theorem
}
\rm
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\centerline{ {\it
Doron 
ZEILBERGER}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg  at math dot rutgers dot edu} ,
\hfill \break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg} .
April 1, 2006.
}
}

{\bf Theorem} (Goldbach [3], see also [1][2][4])
$$
{\sum_{m,n \geq 2}}' {{1} \over {m^n-1}} =1 \quad,
$$
where ${\sum}'$ means that every term only occurs once
(for example $1/15=1/(16-1)$ is only added once even though
$16=4^2=2^4$).

{\bf Proof}: Let $R$ denote the set of all integers larger than $1$ that are {\it not} perfect
powers: $R=\{2,3,5,6,7,10,11,12,13,14,15, \dots \}$. 
Since every  perfect power can be written uniquely as $r^s$ ($r \in R$, $s \geq 2$)
and every  integer $\geq 2$ can be written uniquely as $r^s$ ($r \in R$, $s \geq 1$),
we have
$$
{\sum_{m,n \geq 2}}' {{1} \over {m^n-1}} =
\sum_{r \in R} \sum_{s=2}^{\infty} {{1} \over {r^s-1}} =
\sum_{r \in R} \sum_{s=2}^{\infty} \sum_{i=1}^{\infty} {{1} \over {r^{si}}}=
\sum_{r \in R} \sum_{i=1}^{\infty} \sum_{s=2}^{\infty} ({{1} \over {r^i} })^s=
$$
$$
\sum_{r \in R} \sum_{i=1}^{\infty} 
{
 {
  ({{1} \over {r^i}})^2
 } 
 \over 
 { 
    1- {{1} \over {r^i}}   
 }  
}
=\sum_{r \in R} \sum_{i=1}^{\infty} {{1} \over {r^i(r^i-1)}}
=\sum_{m=2}^{\infty} {{1} \over {m(m-1)}} 
=\sum_{m=2}^{\infty} \left ( {{1} \over {m-1}}-{{1} \over {m}} \right )
=1 \quad . \quad \halmos
$$

{\bf References}

1. E. Catalan, {\it Note sur la sommation de quelques s\'eries},
Journal de Math\'ematiques Pures et Appliqu\'ees {\bf 7} (1842), 1-12.

2. G. Chrystal,``{\it Algebra }'', Part II, reprinted by Chelsea, N.Y. 1964, [p. 422].

3. C. Goldbach, {\it Letter to L. Euler}, 1737.

4. R. L. Graham, O. Patashnik and D.E. Knuth, ``{\it Concrete Mathematics}'',
Addison Wesley, 1989, [p. 66].


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