Automatic Discovery of Irrationality Proofs and Irrationality Measures
By Doron Zeilberger and Wadim Zudilin
.pdf
.ps
.tex
Written: Dec. 20, 2019.
[Appeared in International Journal of Number Theory v. 17 (2021), 815825. (will also appear in a book dedicated to Bruce Berndt)]
A late 80th birthday present to Bruce Berndt (b. March 13, 1939)
Maple packages
Sample Input and Output for ALLADI.txt

If you want to see a computergenerated book about the irrationality of log(1+b/a) for many a and b
(an automatic redux of Theorem 1 in the AlladiRobinson paper)
input file generates the following
computergenerated article.

If you want to see irrationality proofs and measures of 89 constants that came from
int(1/P(x),x=0..1) for searching systematically among all quadratic polynomials P(x)=a+b*x+c*x^2 with
1 ≤ a,b,c, ≤ 10,
the input file generates the following
computergenerated article.

If you want to see irrationality proofs and measures of 43 constants that came from
int(1/P(a+c*x^2),x=0..1) , for a and c between 3 and 40
input file generates the following
computergenerated article.

If you want to see proofs of irrationality and an explicit (as an expression in a) irrationality measure for
arctan(sqrt(a))/sqrt(a) for ALL integers a that are 3(mod 4)
input file generates the following
computergenerated article.
Sample Input and Output for GAT.txt

If you want to see a computergenerated book about the irrationality (and irrationality measure) about the constants
int(1/(1+x/a),x=0..1) for a from 1 to 40, the
input file generates the following
computergenerated book.

If you want to see a computergenerated book about the irrationality (and irrationality measure) about the constants
int(1/(1+x^2/a),x=0..1) for a from 1 to 40, the
input file generates the following
computergenerated book.

If you want to see a computergenerated book about the irrationality (and irrationality measure) about the constants
int(1/(1+x^3/a),x=0..1) for a from 1 to 40, the
input file generates the following
computergenerated book.

If you want to see a computergenerated book about failed attempts at irrationality (and irrationality measure) about the constants
int(1/(1+x^5/a),x=0..1) for a from 1 to 40, the
input file generates the following
computergenerated book.
Input and Output file for BEUKERS.txt

If you want to see a computergenerated paper that handles all the cases for integer linear combinations
of 1, DiLog((a1)/a), and Log((a1)/a) for an ARBITRARY (symbolic!) integer a>=2, the
input file generates the following
computergenerated article.
Input and Output file for CatC.txt

If you want to see a computergenerated paper that treats the attempted relation
of 1, Catalan's constant, Pi, and log(2)
input file generates the following
computergenerated article.
Sample Input and Output for SALIKHOV.txt

If you want to see a computergenerated paper that uses V. Kh. Salikhov's method to discover and prove
(modulo divisibility lemmas left to the reader) linear independence measures of
{1, log(a/(a+1)), log(b/(b+1))}
for many pairs of positive integers 2 ≤ a < b ≤ 100
the input file generates the following
computergenerated article.

If you want to see a computergenerated paper that uses V. Kh. Salikhov's method to discover and prove
(modulo divisibility lemmas left to the reader) linear independence measures for (infinitely many cases!)
{1, log(a/(a+1)), log((a+1)/(a+2))} FOR ALL integers a ≥ 1
the input file generates the following
computergenerated article.

If you want to see a computergenerated paper that uses V. Kh. Salikhov's method to discover and prove
(modulo divisibility lemmas left to the reader) linear independence measures for (infinitely many cases!)
{1, log(a/(a+1)), log((a+1)/(a+2))} FOR ALL integers a ≥ 1
the input file generates the following
computergenerated article.
Sample Input and Output for RUKHADZE.txt

If you want to see a computergenerated paper exploring promising (A,B,C) in the
generalized AlladiRobinson integral
Int((x^{A} (1x)^{B}/(a+x)^C)^{n}/(a+x),x=0..1)
the input file generates the following
output file.

If you want to see computergenerated papers about rigorous crude upper bounds for the irrationality of log((a+1)/a) for
a from 1 to 6, with the original AlladiRobinson case (A,B,C)=1 as well with the
Rukhadze choice (A,B,C)=(6,8,7)
the input file generates the following
output file.
Wadim Zudilin's Home Page
Articles of Doron Zeilberger
Doron Zeilberger's Home Page