Automatic Discovery of Irrationality Proofs and Irrationality Measures

By Doron Zeilberger and Wadim Zudilin

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Written: Dec. 20, 2019.

[To appear in International Journal of Number Theory, and a book dedicated to Bruce Berndt]

A late 80-th birthday present to Bruce Berndt (b. March 13, 1939)

# Maple packages

• ALLADI.txt, a Maple package, inspired by the seminal article "Legendre polynomials and irrationality" by Krishna Alladi and M.L. Robinson, Crelle's J. v. 318 (1980), 137-155. It does an automated redux of Theorem 1 and in their papers, and extends their results treating integrals of the form

int(x^n*(1-x)^n/P(x)^(n+1),x=0..1)

for linear and quadratic polynomials with integer coefficients.

• GAT.txt, a Maple package that includes the former case, but generalizes it to integrals of the form int(1/(1+x^k/a),x=0..1);

• BEUKERS.txt, a Maple package for getting integer linear combinations of 1, DiLog((a-1)/a)), and Log((a-1)/a) that are very small. In particual, if you did not know that Log((a-1)/a)) is irrational, it implies that DiLog((a-1)/a)) and Log((a-1)/a)) can't BOTH be rational.

• CatC.txt, a Maple package for getting (diappointing) integer linear combinations of 1, Catalan, Pi, and log(2) that while are interesting from a numerical-analysis point of view, are diappointing from a number-theoretic. It is not mentioned in the body of the paper.
In particual, even if you did not know before that log(2) and Pi are irrational, it would be impossible to deduce that at least one of {Catalan,Pi,log(2)} are irrational.

• RUKHADZE.txt, a Maple package for exploring E.A. Rukhadze's approach for the irrationality of log(2) (since it was superceded by Marcovecchio), we don't mention it in the paper itself.

• SALIKHOV.txt, a Maple package that uses V. Kh. Salikhov's method to discover and prove (modulo divisibility lemmas left to the reader) linear independence measure of {1, log(a/(a+1)), log(b/(b+1))} for most pairs of integers 2 ≤ a < b

# Sample Input and Output for ALLADI.txt

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

# Sample Input and Output for GAT.txt

• If you want to see a computer-generated 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 computer-generated book.

• If you want to see a computer-generated 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 computer-generated book.

• If you want to see a computer-generated 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 computer-generated book.

• If you want to see a computer-generated 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 computer-generated book.

# Input and Output file for BEUKERS.txt

• If you want to see a computer-generated paper that handles all the cases for integer linear combinations of 1, DiLog((a-1)/a), and Log((a-1)/a) for an ARBITRARY (symbolic!) integer a>=2, the
input file generates the following computer-generated article.

Input and Output file for CatC.txt

• If you want to see a computer-generated paper that treats the attempted relation of 1, Catalan's constant, Pi, and log(2)
input file generates the following computer-generated article.

# Sample Input and Output for SALIKHOV.txt

• If you want to see a computer-generated 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 computer-generated article.

• If you want to see a computer-generated 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 computer-generated article.

• If you want to see a computer-generated 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 computer-generated article.

# Sample Input and Output for RUKHADZE.txt

• If you want to see a computer-generated paper exploring promising (A,B,C) in the generalized Alladi-Robinson integral

Int((xA (1-x)B/(a+x)^C)n/(a+x),x=0..1)

the input file generates the following output file.

• If you want to see computer-generated papers about rigorous crude upper bounds for the irrationality of log((a+1)/a) for a from 1 to 6, with the original Alladi-Robinson 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.