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%a Plain TeX file of an article D. Zeilberger
 
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\bf
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{Research Announcement: The Transcendence of $e+\pi$ and $e \pi$}
\rm
\bigskip
\centerline{ {\it Doron Zeilberger}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
%\break
{\eighttt zeilberg@math.temple.edu} \hfill \break
{\eighttt http://www.math.temple.edu/\Tilde zeilberg   .}
4/1/97.
Submitted to PNAS (via Don Knuth).
} 
}
 
There are $\aleph$ real numbers, while there are only $\aleph_0$
algebraic numbers. Hence the probability that a randomly chosen 
real number is algebraic equals 
${{\aleph_0} \over {\aleph}}={{1} \over {\aleph}}$.
Nevertheless, it is notoriously difficult to prove that 
any {\it naturally occurring} number is irrational, let alone
transcendental.
 
In 1978, Roger Ap\'ery astounded the mathematical
world with his proof of the irrationality of $\zeta(3)$.
In his delightful account, Alf van der Poorten said( [vdP], p. 197):
 
{\it Indeed, proving the irrationality of $\zeta(2n+1)$, $n=2,3, \dots$,
constitutes one of the outstanding problems of the theory
(ranking with the arithmetical nature of 
$\gamma:=\lim_{n \rightarrow \infty} (1+ \dots + (1/n) -\log(n) )$,
and of $e + \pi$, $e \pi$, ..., which are yet undetermined).
}
 
The purpose of this note is to announce that Hermite's[H] celebrated
result that $e$ is transcendental, combined with an amazing
(but apparently overlooked) statement of Goodwin[G], imply
the transcendence of both $e + \pi$ and $e \pi$.
 
But even more interesting than the above implication is the
way by which it was arrived, via {\it computer-generated deduction}.
 
We first developed a 
C-based meta-language, MISPAR, that has built-in number-theoretical
deduction capabilities, that inputs suitably formatted statements
about numbers (especially targeted to handle transcendence theory),
and outputs new statements. Then, using 
ten diligent graduate students,
many results that appeared in papers on the subject were entered
in the appropriate format. Then we used a {\it genetic algorithm} to
deduce million of new results, most of them either trivial or
uninteresting (or both!).
 
Then we made a long list of open problems.
Whenever the computer made a new deduction, it was compared against
the statements in the list, looking for possible matches.
 
While we sure hoped to obtain new {\it interesting} 
results, even in our wildest
dreams we did not anticipate such a spectacular deduction.
 
We are sure that {\it MISPAR} would make many more interesting
deductions in the future. The package itself, and implementation 
details, will be eventually published at the author's website
{\tt http://www.math.temple.edu/\Tilde zeilberg}.
 
{\bf References}
 
[G] E. J. Goodwin, {\it Amer. Math. Monthly}, {\bf 1} (1894),
246-247.
 
[H] C. Hermite, {\it Comptes Rend. Acad. Sci. Paris}, {\bf 77}(1873),
18-24, 74-79, 285-293.
 
[vdP] A. van der Poorten, {\it Math. Intell.}, {\bf 1}(1979), 195-203.
 
 
 
 
 
 
 
 
 
\bye
 
 
