Occam Press  3x+1 Problem, Fermat's Last Theorem, Other Topics in Mathematics

 
3N + 1 Problem
Information about Occam Press and about this web site.
 
3N + 1 Problem The following papers, essays, and notes by Peter Schorer:
Papers on the 3x + 1 Problem (aka the 3n + 1 Problem, the Syracuse Problem, etc.) including:
       "A Solution to the 3x + 1 Problem"
Paper, "Is There a 'Simple' Proof of Fermat's Last Theorem? Several New Approaches"
Paper, "A Challenge for Undergraduate Math Majors"
Paper, "A New Insight Into an Old Calculus Mystery: x, y, dx, dy and the Nature of the Infinitesimal"
Paper, "Occam's Razor and Program Proof By Test"
Paper, "Simulation Paradoxes"
Essay, "Notes on Self-Representing, and Other, Information Structures"
Essay, "A Few Off-the-Beaten-Track Observations and Challenges in Economics, Physics, Computer Science, and Mathematics"  
Essay, "Notes Toward a Pragmatics-Based Linguistics"  
3N + 1 Problem William Curtis's book, How to Improve Your Math Grades, which sets forth a radical new organization of mathematical subjects aimed at improving the speed of problem solving.
3N + 1 Problem Paper, "Good Mathematical Writing Style: Summary of Rules"



Information about Occam Press:

Occam Press is a small publisher located in Berkeley, CA. It was created to provide an outlet for independent scholars, including mathematicians and computer scientists working outside the university.

We will be placing entire works on this web site. Interested persons will be able to buy printed copies directly from us. However, until the works have been placed on the web site, we offer brief descriptions of each. Interested persons may obtain sample pages, and more information, by e-mailing or calling us, or by sending us surface mail.

Occam Press
2538 Milvia St.
Berkeley, CA 94704-2611

E-mail: peteschorer@gmail.com
Tel: (510) 548-3827
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3N + 1 Problem 3N + 1 Problem 3N + 1 Problem 3N + 1 Problem 3N + 1 Problem

Peter Schorer's papers on the 3x + 1 Problem:


"Are We Near a Solution to the 3x + 1 Problem?" -- This paper sets forth several possible strategies for solving the 3x + 1 Problem that are based on the two principal structures that underlie the 3x + 1 function, namely, tuple-sets and recursive "spiral"s. To download a copy as a 211kb PDF file, click here: "Are We Near a Solution to the 3x + 1 Problem?"
Note: This download takes an average of about 1:30 over dial-up connections.

"A Solution to the 3x + 1 Problem" -- To download a copy as a 343kb PDF file, click here: "Solution"
Note: This download takes an average of about 1:50 over dial-up connections.

"Is It Legitimate to Begin a Sentence With 'If Counterexamples Exist, Then...'?" -- To download a copy as a 144kb PDF file, click here: "If Counterexamples Exist..."
Note: This download takes an average of about 0:30 over dial-up connections.

"The Structure of the 3x + 1 Function: An Introduction" -- This paper sets forth an overview of Schorer's main results on the 3x + 1 Problem. It includes definitions, formal statements of all major results, diagrams describing two very simple structures -- tuple-sets and recursive  spirals -- underlying the 3x + 1 function, plus a description of various strategies suggested by these structures.

Section 1. Tuple-sets -- the structure of the function in the "forward" direction, and Section 2. Recursive "spirals" -- the structure of the function in the "backward" direction, i.e., the structure of the inverse of the function. Pages 1-46. To download a copy of Sections 1 and 2 as a 315kb PDF file, click here: "Sections 1 and 2"
Note: This download takes an average of about 1:45 over dial-up connections.

Appendices and Index. Pages 46-98. To download a copy as a 373kb PDF file, click here: "Appendices and Index"
Note: This download takes an average of about 2:05 over dial-up connections.

"The Structure of the 3x + 1 Function" -- Contains proofs of all theorems and lemmas not proved in the above papers. Extensive table of contents and index.

Section 1. Tuple-sets -- the structure of the function in the "forward" direction. Pages 1-62. To download a copy as a 515kb PDF file, click here: "Section 1. Tuple-sets"
Note: This download takes an average of about 2:50 over dial-up connections.

Section 2. Recursive "spirals" -- the structure of the function in the "backward" direction, i.e., the structure of the inverse of the function. Pages 63-107. To download a copy as a 310kb PDF file, click here: "Section 2. Recursive 'spirals'"
Note: This download takes an average of about 1:40 over dial-up connections.

To access a web site with an excellent overview of some recent results on the 3x + 1 Problem, click here.

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About Peter Schorer's paper, "Is There a 'Simple' Proof of Fermat's Last Theorem? Several New Approaches":

Finding a proof of Fermat's Last Theorem was the most famous unsolved problem in mathematics until Andrew Wiles discovered a proof in the mid-1990s -- one that was several hundred pages in length (actually, Wiles proved part of the Shimura-Taniyama Conjecture, which implies the truth of the Theorem). Nevertheless, the question remains, was Fermat really correct, when he wrote, in the 1600s, in his copy of Diophantus' book on number theory, that "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain"? If he was correct, then, in comparison to Wiles' sophisticated modern proof, his can be called "simple" if for no other reason than that he accomplished it without any of the advanced theory that Wiles used.

Or there may be other "simple" proofs that differ from Fermat's. The question will probably always be of interest to a few mathematicians, at least until someone (or some theorem-proving computer program) finds such a proof or until one or more programs give compelling evidence that it is highly unlikely that such a proof exists.

In his paper, Schorer explores several possible approaches to a simple proof.

Part (1) describes the approaches.

To download a copy as a 697kb PDF file (65 pages), click here:
"...'Simple' Proof (1)"
Note: This download takes an average of about 3:00 over dial-up connections.

Part (2) contains proofs of lemmas not proved in Part (1).

To download a copy as a 342kb PDF file (24 pages), click here:
"...'Simple' Proof (2)"
Note: This download takes an average of about 1:30 over dial-up connections.

Part (3) contains a description of failed implementations of some of the ideas in Part (1).

To download a copy as a 119kb PDF file (5 pages), click here:
"...'Simple' Proof (3)"
Note: This download takes an average of about 0:45 over dial-up connections.

Part (4) contains details on approaches to a proof of FLT via the "lines-and-circles" model of congruence.

To download a copy as a 346kb PDF file (24 pages), click here:
"...'Simple' Proof (4)"
Note: This download takes an average of about 1:30 over dial-up connections.


"A Challenge for Undergraduate Math Majors" -- Schorer has written up one of his strategies for a simple proof of Fermat's Last Theorem in a self-contained paper in which he challenges undergraduate math majors to find an error, and if possible, to fix it. To download a copy as a 165kb PDF file, click here: "A Challenge..."
Note: This download takes an average of about 1:00 over dial-up connections.

"A New Insight Into an Old Calculus Mystery: x, y, dx, dy and the Nature of the Infinitesimal". To download a copy as a 163kb PDF file, click here: "A New Insight..."
Note: This download takes an average of about 1:00 over dial-up connections.

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About Peter Schorer's paper on the validity of Occam's Razor in a program testing context:

The point of departure of Peter Schorer's paper, "Occam's Razor and Program Proof By Test," is the following question: suppose we have two computer programs, one long and one short (where "long" and "short" refer to the number of instructions in each program), and both programs test correct for some finite number of inputs. Which program is more likely to be correct for all inputs (where "all" means an infinite number)?

The principle of intellectual economy known as "Occam's Razor" says that the shorter program is more likely to be correct for all inputs. But can we prove this?

The answer is yes if the programs happen to be the equivalent of what are called "finite-state machines", i.e., programs with only a finite amount of memory. Furthermore, the answer is yes for a certain limited class of programs that have unlimited memory, but in this case, the number of nested loop statements is severely limited. (Loop statements are statements of the form, "while such-and-such is true do begin ... end" or "for all numbers in the range such-and-such do begin ... end" . Nested loop statements are loop statements that contain loop statements inside the "begin ... end" part.)

One reason why this limitation is present is that the class of program allows what Schorer calls "elusive errors" -- errors which can occur unpredictably, e.g., "if there are three successive 7s in the decimal expansion of pi beginning at the nth digit of pi (where n is the input to the program) then output the following erroneous value...". Suppose we don't allow elusive errors in any program in the class. Can we then allow these programs to have an arbitrary number of nested loops and still be assured that a finite number of tests will reveal if the program is correct for all inputs?

Schorer describes a class of program which cannot have elusive errors, then presents strong plausibility arguments why finite testing of correctness is possible in this class.

The paper has an extensive table of contents and index. No. of pages: 25.

This paper will be included in the forthcoming revision of Schorer's 1985 cult classic,
Shaving With Occam's Razor.

To download a copy as a 229 kb PDF file, click here: "Occam's Razor and Program Proof By Test"
Note: This download takes an average of about 1:10 over dial-up connections.

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About Peter Schorer's paper, "Simulation Paradoxes":

This paper attempts to explain Newcomb's Paradox and the Paradox of the Unexpected Hanging by showing that they arise from reasoning processes that mutually simulate each other.

To download a copy as a 31kb PDF file, click here: "Simulation Paradoxes."
Note: This download takes an average of about 0:20 over dial-up connections.

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About Peter Schorer's Essay, "Notes on Self-Representing, and Other, Information Structures":

In this essay, Schorer first defines "information structure", then considers three types: (1) those having "too little" information (examples include approximations in mathematics, computer science, and graphics);

(2) those having "enough" information -- so-called "self-representing" or "light" information structures (any structure which can contain a description of the entire structure at each of its nodes; thus, e.g., subway systems in which it is possible to have a map of the entire system at each station; airplanes that can carry all the drawings and specifications that describe them); and

(3) those structures having "too much" information (examples include several paradoxes which arise from the "superposition" of conflicting information).

To download a copy as a 335kb PDF file, click here: "Self-Representing, and Other, Information Structures."
Note: This download takes an average of about 1:50 over dial-up connections.

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About Peter Schorer's "A Few Off-the-Beaten-Track Observations and Challenges in Economics, Physics, Computer Science, and Mathematics":

This is a chapter in the forthcoming revision of Schorer's book, Shaving with Occam's Razor.  The reader can get an idea of the range of subject matter from the table of contents in each of two PDF files. The first file and first part of the chapter addresses economics, physics, and computer science and the second, mathematics.

To download a copy of the first part as a 199kb, 25 page PDF file, click here: "A Few Off-the-Beaten-Track Observations and Challenges in Economics, Physics, Computer Science, and Mathematics: Economics, Physics, and Computer Science"
Note: This download takes an average of about 1:20 over dial-up connections.

To download a copy of the second part as a 288kb, 35 page PDF file, click here: "A Few Off-the-Beaten-Track Observations and Challenges in Economics, Physics, Computer Science, and Mathematics: Mathematics"
Note: This download takes an average of about 1:35 over dial-up connections.

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About Peter Schorer's Essay "Notes Toward a Pragmatics-Based Linguistics":

In this short paper, Schorer sets forth an argument that syntax and semantics can be derived from pragmatics (the study of the circumstances in which sentences, etc., occur in a language). He also argues that one of the most important characteristics of a language is the frequency-of-occurrence of words, phrases, sentences, etc.

Some of the ideas in this essay are derived from the chapter "Language" in John Franklin's book Thoughts and Visions on the web site www.thoughtsandvisions.com

To download a copy as a 149kb, 8 page PDF file, click here: "Notes Toward a Pragmatics-Based Linguistics"
Note: This download takes an average of about 1:00 over dial-up connections.

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About William Curtis' How to Improve Your Math Grades:

This book sets forth a new method for students to organize their notes for any math course (in fact, for any technical course), a method that is optimized for speed of problem solving.  Instead of the typical linear structure of a subject--definitions, theorems or lemmas, proofs, definitions, theorems or lemmas, proofs...--this method shows students how to organize notes alphabetically by concept or mathematical entity.  (An entity might be, e.g., a type of number or function or set or...)

It then describes a "template" for organizing all content pertaining to that entity, a template that includes:

  • definition of the entity;
  • ways of representing the entity;
  • common operations on the entity:
    • determining if two instances of the entity are equal;
    • performing arithmetic operations on the entity (if appropriate);
    • creating more of the entity;
    • breaking down a given instance of the entity into fundamental building blocks; etc.
  • theorems on the entity;
  • types of the entity;
  • related entities, concepts
Thus, for example, in high school algebra, the entities include integers, fractions, irrational numbers, functions, equations. For equations, the common operations include:
  • determining if an equation has any solutions, and, if so, how many;
  • solving an equation;
  • converting a given equation into a given canonical form, e.g., ax2 + bx + c = 0;
    • getting the unknown on one side and everything else on the other;
    • multiplying or dividing all terms of an equation by some term;
    • moving terms from one side of the equation to the other;
    • adding and subtracting equations; etc.
The book contains numerous examples of this new organization of notes applied to subjects in undergraduate mathematics, e.g., number theory.  It also describes a new method of doing proofs called "structured proof," and gives examples.  This method makes much easier the understanding of existing proofs, and the discovery of proofs required in homework and examination problems.

The book is a generalization of ideas in Peter Schorer's "How to Create Zero-Search-Time Computer Documentation", which is accessible online at ZSTHelp.com/book

Total no. of pages: 258.  Preface, chapters, appendices and index are individually downloadable as PDF files as follows:

To contact the author click here

Title page, etc.
To download a copy of the Title page, etc. as a 107kb PDF file, click here: Title page, etc.
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Preface, pages i-ii
To download a copy of the Preface as a 94kb PDF file, click here:  Preface
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Chapter 1  Why Is Mathematics Difficult? pages 1-11
To download a copy of Chapter 1 as a 152kb PDF file, click here:  Chapter 1
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Chapter 2  Mathematics in the University, pages 12-40
To download a copy of Chapter 2 as a 267kb PDF file, click here:  Chapter 2
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Chapter 3  Fundamental Concepts, pages 41-62
To download a copy of Chapter 3 as a 219kb PDF file, click here:  Chapter 3
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Chapter 4  How to Build an Environment, pages 63-96
To download a copy of Chapter 4 as a 300kb PDF file, click here:  Chapter 4
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Chapter 5  Proofs, pages 97-115
To download a copy of Chapter 5 as a 213kb PDF file, click here:  Chapter 5
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Chapter 6  More Ideas to Help You Build Better Environments (part 1), pages 116-148
To download a copy of Chapter 6 (part 1) as a 290kb PDF file, click here: Chapter 6 (part 1)
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Chapter 6  More Ideas to Help You Build Better Environments (part 2), pages 149-168
To download a copy of Chapter 6 (part 2) as a 222kb PDF file, click here: Chapter 6 (part 2)
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Chapter 7  Homework and Exams, pages 169-173
To download a copy of Chapter 7 as a 97kb PDF file, click here:  Chapter 7
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Chapter 8  For Future Mathematicians Only, pages 174-200
To download a copy of Chapter 8 as a 243kb PDF file, click here:  Chapter 8
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General Remarks on the Appendices, pages 201-202
To download a copy of General Remarks on the Appendices as a 67kb PDF
file, click here:  General Remarks on the Appendices
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A Number Theory Environment (partial), pages 203-217
To download a copy of A Number Theory Environment as an 219kb PDF file, click here:
A Number Theory Environment
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Example of Using an Environment to Solve a Problem, pages 218-222
To download a copy of Example of Using an Environment to Solve a Problem as a 114kb PDF file, click here:
Example of Using...
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Linear Algebra: Partial KWIC Index of Definitions, Lemmas and Theorems
To download a copy of Linear Algebra: Partial KWIC Index of Definitions, Lemmas and Theorems as a 150kb PDF file, click here:
Linear Algebra...
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Global-View and Ease-of-Understanding Hierarchies, pages 223-234
To download a copy of Global-View...Heirarchies as a 150kb PDF file, click here:
Global-View...Heirarchies
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Index, pages 235-258
To download a copy of the Index as a 681kb PDF file, click here:  Index
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Good Mathematical Writing Style: Summary of Rules


To download a copy as a 134kb PDF file, click here: "Good Mathematical Writing Style"
Note: This download takes an average of about 0:45 over dial-up connections.

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