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### Automated conjecture making in number theory using HR, Otter and Maple

Simon Colton
2005 Journal of symbolic computation
We add some automation to this discovery process by using the HR theory formation system to make conjectures about Maple functions supplied by the user.  ...  We describe the core functionality of HR which enables it to form a theory, and the additional functionality implemented in order for HR to work with Maple functions.  ...  We would also like to thank the anonymous referees from the Calculemus conference for their interesting comments and suggestions about this work, and the organisers of that conference, at which this material  ...

### Automated Proofs of Many Conjectured Recurrences in the OEIS made by R.J. Mathar [article]

Shalosh B. Ekhad, Mingjia Yang, Doron Zeilberger
2017 arXiv   pre-print
In this case study we demonstrate, and actually fully implement (in an accompanying Maple package), how to turn many conjectures made in the OEIS by R.J.  ...  ", as opposed to "proved conjecture" (alias theorem).  ...  The Maple package SCHUTZENBERGER.txt While many of the needed functions can be found in the Maple package gfun [SZ] mentioned above, we found it more convenient to write our own Maple code, SCHUTZENBERGER.txt  ...

### On Euler's "Misleading Induction", Andrews' "Fix", and How to Fully Automate them [article]

2013 arXiv   pre-print
Usually he had an uncanny intuition on how many "special cases" one needs before one can formulate a plausible conjecture, but one time he was "almost fooled", only to find out that his conjecture was  ...  But in order to turn the empirical proof into a full-fledged rigorous proof, we must make sure that we check sufficiently many (but still not that many!) special cases.  ...  Interlude: Even Giants make stupid conjectures We are a little surprised that Euler could have believed, even for a second, that Eq. (Leonhard) is true for all n.  ...

### Liebe Opa Paul, ich bin auch ein experimental Scientist!

Doron Zeilberger
B 22 (n, r + 2)B 11 (n, r + 1) = 1, which Maple (or Mathematica, etc.) can verify routinely, and hence prove the conjecture.  ...  He also talked about Algorithmic Paradigms. In particular, about Hierarchy, Adaptivity, and (De)composition.  ...

### The Holonomic Ansatz II. Automatic Discovery(!) And Proof(!!) of Holonomic Determinant Evaluations

Doron Zeilberger
2007 Annals of Combinatorics
The main functions are Rproof, RproofP, SRproof, SRproofI, to find out about them, in DET, type ezra(FunctionName);. For example, with help for Rproof, type ezra(Rproof);.  ...  Let me just mention that DaveH works by conjecturing recurrences for many rows (i.e. for fixed n), and then using the procedure GR1 that guesses rational functions in order to guess a unified recurrence  ...

### Disturbing the Dyson Conjecture (in a GOOD Way)

Andrew V. Sills, Doron Zeilberger
2006 Experimental Mathematics
We present a case study in experimental yet rigorous mathematics by describing an algorithm, fully implemented in both Mathematica and Maple, that automatically conjectures, and then automatically proves  ...  , closed-form expressions extending Dyson's celebrated constant term conjecture.  ...  /Maple needs to supply a conjecture.  ...

### Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type [article]

2011 arXiv   pre-print
Monthly Problem, to prove a nice explicit formula for the generating function for the number of n-letter words in H,T that have as many occurrences of HT as HH.  ...  What About Several Distinguished Words?  ...  But there is another way to make everything fully rigorous. Via the holonomic ansatz!  ...

### Explicit expressions for the moments of the size of an (n, dn-1)-core partition with distinct parts [article]

Anthony Zaleski
2017 arXiv   pre-print
We exhibit formulas for the moments of the size, as functions of d with n fixed, and vice versa. We conjecture that the distribution is asymptotically normal as n approaches infinity.  ...  In particular, we conjectured that the distribution is asymptotically normal.  ...  Then X d,n is a random variable, so it makes sense to inquire about its distribution.  ...

### A proof of Julian West's conjecture that the number of two-stacksortable permutations of length n is 2(3n)!/((n + 1)!(2n + 1)!)

Doron Zeilberger
1992 Discrete Mathematics
expected, and required about 50 mathematician-hours, 10 (Maple) programmerhours, (the mathematician and programmer being myself), and 10 CPU-hours to consfruct the proof.  ...  A naive approach is to 'solve' F( Y, x, t) 'explicitly', say by radicals, and verify that it satisfies the functional equation G = 0 of Step 1. However, Maple was unable to do it.  ...  Appendix: A Maple program for Steps 3 and 4 # this is the functional equation of step 1, derived humanly in # Section 2 G := PHI -l/(1 -x*t) -x*t*(P -t*PHI)*(P -PHI)/(lt)^2: G := normal (G): G := numer  ...

### Rademacher's infinite partial fraction conjecture is (almost certainly) false

Andrew V. Sills, Doron Zeilberger
2013 Journal of difference equations and applications (Print)
function for partitions of integers into at most N parts exist and equal particular values that he specified.  ...  In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating  ...  after about n = 150.  ...

### Integrals Involving Rudin–Shapiro Polynomials and Sketch of a Proof of Saffari's Conjecture [chapter]

2017 Analytic Number Theory, Modular Forms and q-Hypergeometric Series
Equivalently, the generating function R 1 (t) is given by  ...  Preface: Krishna Alladi One of the greatest disciples of Srinivasa Ramanujan, who did so much to make him a household name in the mathematical community, and far beyond, is Krishnaswami "Krishna" Alladi  ...  Added May 27, 2016: Brad Rodgers, independently, and simultaneously, found a (complete) proof of Saffari's conjecture, that he is writing up now and will soon post in the arxiv.  ...

### A Symbolic Finite-state approach for Automated Proving of Theorems in Combinatorial Game Theory [article]

Thotsaporn "Aek" Thanatipanonda, Doron Zeilberger
2007 arXiv   pre-print
We develop a finite-state automata approach, implemented in a Maple package ToadsAndFrogs available from our websites, for conjecturing, and then rigorously proving, values for large families of positions  ...  In particular, we prove a conjecture of Jeff Erickson.  ...  Then he improved the program by making use of the symbolic computation capability of Maple, to formulate conjectures, and prove the values of game-positions.  ...

### Dave Robbins' art of guessing

Doron Zeilberger
Luckily, the unclassified ten percent of Dave's work is more than enough to make him immortal.  ...  Once we have the notion of ASM, it is not hard, with Maple once again, to conjecture (DaveHoward).  ...  Using classical stuff about Schur functions, one easily gets that the number of Gelfand-Zeitlin patterns with bottom row equaling 12 . . . n is 2 n(n−1)/2 .  ...

### Using Technology in a Differential Equations Course: Lessons Learned Implementing a New Paradigm

Thomas Wangler
2011 CODEE Journal
In an attempt to make the course more interesting and less daunting, a new paradigm was adopted-a paradigm that incorporates technology as a core component of the course, rather than just an "add on. "  ...  Based on the direction field, make a conjecture about the asymptotic behavior of the solutions.  ...  You will need to insert the points [ Based on your graph, make a conjecture about the relationship between the optimal harvesting rate and the initial sardine biomass.  ...