Searching for ApéryStyle Miracles
[Using, InterAlia, the Amazing AlmkvistZeilberger Algorithm]
By
Shalosh B. Ekhad and Doron Zeilberger
.pdf
.ps
.tex
First Written: May 17, 2014
[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org]
Dedicated to Gert Almkvist (b. April 17, 1934), on becoming (at least, chronologically), an octogenerian (plus one month).
[Added Dec. 19, 2019: Sadly Gert Almkvist died on Nov. 24, 2018. He was one of the most original mathematicians that I have known]
Older mathematicians make beautiful provers! One of my great heroes is
Roger Apéry who at
the age of 64 astounded the mathematical world with a proof that
1/1^{3}+1/2^{3}+1/3^{3}+ 1/4^{3}+ ...
is not a ratio of two integers. Since very soon I will turn 64 myself, I was hoping to do
the analogous thing for
1/1^{5}+1/2^{5}+1/3^{5}+ 1/4^{5}+ ... .
Alas, since I am not nearly as smart as Apéry, my only hope was to teach
to Shalosh Apéry's neat tricks, but to our disappointment we could not do it.
Nevertheless, once I taught Shalosh Roger's tricks, it reproduced in one second the
original proof (see below), and found lots and lots of new irrationality proofs of many
new constants, but none of them happened to be ζ(5), or any of the other `famous',
yettobeprovenirrational, constants.
But maybe YOU would be able to use the present Maple package, or
an extension of it, to do it. Good luck!
This article is dedicated to yet another hero of mine, GERT ALMKVIST, who has just turned 80.
Happy Birthday Gert, and may you continue to do great experimental math!
Maple Package
Input and Output files

If you want to see all polynomials of the form
a/x+b+cx
where a,b,c, are integers between 1 and 15, that should (judging from empirical deltas) yield Aperymiracles for
for contants that come from the sequence of constant terms of (a/x+b+cx)^n,
the input yields
the output
Note: In this output file the successful set is named Tovim.

If you want to find out the disappointing fact that no polynomial of the form
a/x^2+b/x+c+dx+ex^2
for 1 ≤ a,b,c,d,e ≤ 5
yields (empirically) Aperystyle miracles
the input yields
the output
Note that the output is the empty set.

If you want to see sequences of empirical deltas, for various ratios of sequences
arising from constant terms (using the amazing AlmkvistZeilberger algorithm)
the input yields
the output
Note that for these examples they are all larger than 1, indicating miracles!

If you want to see (using, again, the amazing AlmkvistZeilberger algorithm),
empirical deltas for sequences of rational numbers converging
to various constants, given by Beukersstyle integral representations, the
the input yields
the output
Note: only the last one, for arctan(1/5), produces an Aperymiracle, but
the other ones are intersting too.

If you want to see very verbose versions of the above, spelling out the
actual rational approximations to the designated constants,
the input yields
the output

If you want to see (using, now, the famous Zeilberger algorithm),
empirical deltas for sequences of rational numbers converging
to various constants, arising from binomial coefficients sum, like
in the original Apéry proofs
the input yields
the output
Note: The first three are the famous Apery ones for log(2), Zeta(2), and Zeta(3),
the fourth (disappointing) and fifth (good!) are random tries.

If you want to see all polynomials of the form
a/x+b+cx
where a,b,c, are integers between 1 and 15, that PROVABLY (rigorously) yield Aperymiracles for
for contants that come from the sequence of constant terms of (a/x+b+cx)^n,
the input yields
the output
Note: In this output file the successful set is named Tovim.

If you want to see all polynomials of the form
a/x+b+cx+dx^2
where a,b,c,d are integers between 1 and 14, that PROVABLY (rigorously) yield Aperymiracles for
for contants that come from the sequence of constant terms of (a/x+b+cx+dx^2)^n,
the input yields
the output
Note: In this output file the successful set is named Tovim.

If you want to find out the disappointing fact that no polynomial of the form
a/x^2+b/x+c+dx+ex^2
for 1 ≤ a,b,c,d,e ≤ 8
yields (rigorously) Aperystyle miracles
the input yields
the output
Note that the output is the empty set.

If you want to see terse versions of fullyrigorous Aperystyle proofs that
come from constantterm expressions using the Amazing AlmkvistZeilberger algorithm
the input yields
the output

If you want to see VERBOSE versions (HUMANLY READABLE ARTICLES!) of fullyrigorous Aperystyle proofs that
come from constantterm expressions using the Amazing AlmkvistZeilberger algorithm
the input yields
the output
Note: the last two sequences come from THIRDorder recurrences. Wow! So Aperystyle
miracles are not confined to sequences satisfying secondorder recurrences.

If you want to see fullyrigorous Beukersstyle proofs using Integral represenations
(but for single integrals rather than double or triple integrals), given in
tersestyle,
the input yields
the output
Note: only the last one yields irrationality, since there the (rigorous) delta is larger than 1.

If you want to see verbose versions of these, with fully humanreadable articles,
the input yields
the output

If you want to see "virgin" data that only come from a linear recurrence
with polynomial coefficients and initial values,
the input yields
the output

If you want to see terse versions of Apery's famous proofs of the irrationality
of log(2), Zeta(2), and Zeta(3)
the input yields
the output

If you want to see verbose versions of Apery's famous proofs of the irrationality
of log(2), Zeta(2), and Zeta(3), i.e. fulllength humanlyreadable proofs,
the input yields
the output

If you want to see symbolic formulas for irrationality measure for families
of constants, for various examples discussed in the article
the input yields
the output

If you want to see symbolic formulas for irrationality measure for families
of constants, for various examples discussed in the article
the input yields
the output

If you want to see the input file for the symbolic formulas for irrationality measure for a
3parameter constant, that yields a thirdorder recurrence equation. Here it is:
input
Unfortunately it took too long, so we don't have an output file.

If you want to see 144 potential Apéry miracles
the input yields
the output

If you want to see why the "first impressions" of Alf van der Poorten in his beautiful article:
(Math. Intell. v. 1 (1979), 195203), footnote 9, p. 202 were wrong`):
the input yields
the output

If you want to see several examples of procedure RAaz NOT discussed in the present article (out of sheer laziness):
the input yields
the output
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