Information about Occam Press and about this web site.
A note to professional mathematicians.
A note to graduate students.
"A Solution to the 3x + 1 Problem"
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Information about Occam Press and about this web site. |
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A note to professional mathematicians. |
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A note to graduate students. |
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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" |
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| Paper, "Is There a 'Simple' Proof of Fermat's Last Theorem? Several New Approaches" | |
| Paper, "Occam's Razor and Program Proof By Test" | |
| Paper, "Simulation Paradoxes" | |
| Essay, "Notes on Self-Representing, and Other, Information Structures" | |
| "A Few Off-the-Beaten-Track Observations and Challenges in Economics, Physics, Computer Science, and Mathematics" | |
| Essay, "Notes Toward a Pragmatics-Based Linguistics" | |
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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. |
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Paper, "Good Mathematical Writing Style: Summary of Rules" |
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@cs.com
Tel: (510) 548-3827
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It is completely understandable that the professional mathematician should be skeptical of the contents of a web site that claims a possible solution to the 3x + 1 Problem, and that contains a paper on the possibility of a simple proof of Fermat's Last Theorem.
Let me give every assurance that I am not a crackpot. (My degree is in computer science, and I was formerly a researcher at the main research laboratory of one of the world's leading computer manufacturers.) The 3x + 1 paper has been read by graduate students and professional mathematicians. All known errors have been repaired. In addition, the paper receives more than 80 visits a month on this web site. No errors have been reported to me.
The skeptical mathematician is invited to improve upon the current implementations of the 3x + 1 strategy. As he or she will see, the strategy is challenging indeed. I will be glad to offer shared authorship in return for an improvement that results directly in publication.
The possible solution to the Problem seemed to suggest an approach to a simple proof of Fermat's Last Theorem, and so I have written a discussion of that approach and of several others derived from it. I make no claims of a proof.
– Peter Schorer
Occam Press
2538 Milvia St.
Berkeley, CA 94704-2611
E-mail: peteschorer@cs.com
Tel: (510) 548-3827
Back to the top
It is completely understandable that you should be skeptical of the contents of a web site that claims a possible solution to the 3x + 1 Problem, and that contains a paper on the possibility of a simple proof of Fermat's Last Theorem.
Let me give every assurance that I am not a crackpot. (My degree is in computer science, and I was formerly a researcher at the main research laboratory of one of the world's leading computer manufacturers.) The 3x + 1 paper has been read by graduate students and professional mathematicians. All known errors have been repaired. In addition, the paper receives more than 80 visits a month on this web site. No errors have been reported to me.
Although I believe I have solved the 3x + 1 Problem, the proofs offered can certainly be improved. There are possibilities of additional proofs in the paper: "Are We Near a Solution to the 3x + 1 Problem?" I will be glad to offer shared authorship to a qualified graduate student who helps me produce a paper that is accepted for publication. I will be glad to pay any reasonable hourly fee for the time you spend, and, of course, I guarantee complete confidentiality. No one will know you worked with me on the Problem unless you tell them. (I can furnish references as to my trustworthiness.) You would be able to set a limit to the amount of time you spend each week. You (or I) would be free to bow out of the agreement at any time.
– Peter Schorer
Occam Press
2538 Milvia St.
Berkeley, CA 94704-2611
E-mail: peteschorer@cs.com
Tel: (510) 548-3827
Back to the top
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Schorer's stategies for a solution of the 3x + 1 Problem are based the following facts, which are not disputed:
(1) The set of all finite iterations of the 3x + 1 function can be partitioned into an infinite set of "tuple-sets";
(2) Each finite tuple is a prefix of an infinite tuple;
(3) In each i-level tuple-set, i ≥ 2, there is exactly one i-level first tuple (anchor tuple);
(4) If counterexamples exist, then each tuple-set contains an infinity of non-counterexample tuples and an infinity of counterexample tuples (Lemma 5.0).
These facts are used to show that (informally) there is not enough "room" for all the anchor tuples that are required by (4).
"Are We Near a Solution to the 3x + 1 Problem?" --
This paper sets forth a possible strategy for solving the 3x + 1 Problem based on tuple-sets, one of two structures that underlie the 3x + 1 function. 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 Brief Description of a Solution to the 3x + 1 Problem" --
To download a copy as a 156kb PDF file, click here: "Sketch"
Note: This download takes an average of about 0:45 over dial-up connections.
"A Solution to the 3x + 1 Problem" --
To download a copy as a 343kb PDF file, click here: "Solution"
The author will welcome the reader's comments, suggestions, and criticisms of this paper.
If you believe this paper should be published, please see Appendix C.
Note: This download takes an average of about 1:50 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 underlying the 3x + 1 function,
plus a description of various strategies suggested by these structures.
The "Pushing-Away" Strategy seems the most promising. In this strategy, tuples representing iterations on counterexamples to the 3x + 1 Conjecture are shown to be "pushed away" from the set of tuples representing iterations on non-counterexamples.
Several possible proofs based on this strategy are given.
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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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.
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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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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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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:
Thus, for example, in high school algebra, the entities include integers, fractions, irrational numbers, functions, equations. For equations, the common operations include:
- 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
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.
- 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 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.
Note: This download takes an average of about 0:30 over dial-up connections.
Preface, pages i-ii
To download a copy of the Preface as a 94kb PDF file, click here: Preface
Note: This download takes an average of about 0:30 over dial-up connections.
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
Note: This download takes an average of about 0:45 over dial-up connections.
Chapter 2 Mathematics in the Schools, pages 12-40
To download a copy of Chapter 2 as a 267kb PDF file, click here: Chapter 2
Note: This download takes an average of about 1:15 over dial-up connections.
Chapter 3 Fundamental Concepts, pages 41-62
To download a copy of Chapter 3 as a 219kb PDF file, click here: Chapter 3
Note: This download takes an average of about 1:00 over dial-up connections.
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
Note: This download takes an average of about 1:30 over dial-up connections.
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)
Note: This download takes an average of about 1:30 over dial-up connections.
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
Note: This download takes an average of about 0:30 over dial-up connections.
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
Note: This download takes an average of about 1:15 over dial-up connections.
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
Note: This download takes an average of about 0:30 over dial-up connections.
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...
Note: This download takes an average of about 0:30 over dial-up connections.
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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