Written: Dec. 20, 2019.

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.
  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.

• 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

• SALIKHOVpi.txt, a Maple package that uses V. Kh. Salikhov's integral (in his paper "On the measure of irrationality of Pi" Math. Notes 88(4) (2010) 563-573, but a different approach using recurrences obtained via the Almkvist-Zeilberger algorithm, to find diophantine approxrimations to Pi that lead to good irrationality measures.

  Note that you also need to download the following data file SalikhovDataFile.txt, and put it in the same directory in your computer.

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 8) or 7 (mod 8)
  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.

• 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.

Sample Input and Output for SALIKHOVpi.txt

• If you want to see a computer-generated paper that does the (2,3)-analog of Salikhov's proof, with a seemingly better irrationality measure (that still needs to be confirmed rigorously)
  the input file generates the following computer-generated article.

• If you want to see a computer-generated paper that redoes, using our approach, the original (3,5) case
  the input file generates the following computer-generated article.

• If you want to see a computer-generated paper that handles again the case (2,3), similar to the one above, but taking advantage of the fact (unlike the original (3,5) case where the odd n yield approximations of arctan(1/7)), that the corresponding Salikhov integral gives combinations of 1 and Pi for any n, hence the recurrence is simpler, but otherwise it is the same,
  the input file generates the following computer-generated article.

Wadim Zudilin's Home Page

Articles of Doron Zeilberger

Doron Zeilberger's Home Page