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Papers by Peter Schorer

Research paper thumbnail of A Very Short Proof of Fermat's Last Theorem

Research paper thumbnail of Is a solution to the 3x + 1 problem in sight?

SIGACT news, Dec 1, 1997

Dec. 14, 2024 "Very often in mathematics the crucial problem is to recognize and to discover what... more Dec. 14, 2024 "Very often in mathematics the crucial problem is to recognize and to discover what are the relevant concepts; once this is accomplished the job may be more than half done." 1 "One of the greatest contributions a mathematician can make is to spot something so simple and powerful that everybody else has missed it." 2 Note 1 : Our proofs of the 3x + 1 Conjecture begin under "Proofs of the 3x + 1 Conjecture " on page 3. (A proof of the Conjecture solves the 3x + 1 Problem.) The reader should begin reading on page 2. Note 2: We will pay 75,000tothenumbertheoristwhohelpsusprepareaversionofthispaperthatresultsinourwlnningthenearly75,000 to the number theorist who helps us prepare a version of this paper that results in our wlnning the nearly 75,000tothenumbertheoristwhohelpsusprepareaversionofthispaperthatresultsinourwlnningthenearly1 million prize for a solution to the 3x + 1 problem that is being offered by a Japanese organization. See details in Appendix D at the end of this paper.

Research paper thumbnail of A Solution to the 3x + 1 Problem

The 3x + 1 Problem asks if repeated iterations of the function C(x) = (3x + 1)/(2a) always termin... more The 3x + 1 Problem asks if repeated iterations of the function C(x) = (3x + 1)/(2a) always terminate in 1 Here x is an odd, positive integer, and a = ord2(3x + 1), the largest positive integer such that the denominator divides the numerator. The conjecture that the function always eventually terminates in 1 is the 3x + 1 Conjecture. An odd, positive integer that maps to 1 is called a non-counterexample; an odd, positive integer that doesn’t map to 1 is called a counterexample (to the Conjecture). Our first proof (given under:“First Proof of the 3x + 1 Conjecture” on page 11) is based on a structure called tuple-sets that represents the 3x + 1 function in the “forward” direction. In our proof, we show that the 35-level elements of all 35-level tuples in all 35-level tuple-sets are the same, regardless if counterexamples to the Conjecture exist or not1. From this fact, a simple inductive argument allows us to conclude that all tuple-sets are the same, whether counterexamples exist or not, and hence that counterexamples do not exist. Our second proof (given in “Second Proof of the 3x + 1 Conjecture” on page 12), like the first, is based on tuple-sets. In this proof, we define anchor, which is the i-level element of the first i-level tuple in an i-level tuple-set. We then show that there is one and only one set of anchors for all i, regardless if counterexamples exist or not. We then show that this implies that there is one and only one set of infinite tuples, regardless if counterexamples exist or not, and from this we deduce that, if counterexamples exist, then some infinite tuples must be both counterexample and non-counterexample tuples, which is absurd, hence counterexamples do not exist and the Conjecture is true. Our third proof (given in “Third Proof of the 3x + 1 Conjecture” on page 21) has three versions. The first is based on a structure called the 1-tree. This tree is a y-tree, where the root y is a range element of the 3x + 1 function; y-trees represent the 3x + 1 function in the “inverse” direction. The second version is based on the remarkable fact that each finite sequence of iterations of the 3x + 1 function can in principle be traced on a certain spiral diagram. The third version is also based on the 1-tree. As far as we have been able to determine, our approaches to a solution of the Problem are original. 1. A phrase of the form “q regardless if p” is equivalent to “(if p then q) and (if not-p then q)”. It is meaningful and in fact true as long as q is true, which it always is in this paper. Instances of the phrase occur in everyday speech, for example, “Fermat’s Last Theorem is true regardless if the Riemann Conjecture is true”.,

Research paper thumbnail of Structure the Use: Notes on a Method for Designing Computing System Environments

IEEE Computer, Dec 1, 1981

We may be able to design a better computer system if we think of it as an entity inseparable from... more We may be able to design a better computer system if we think of it as an entity inseparable from all of the training, documentation, and folklore that help make it work.

Research paper thumbnail of A Discussion of Several Possible Strategies

Readers can safely assume, initially, that all referenced lemmas are true, since their proofs hav... more Readers can safely assume, initially, that all referenced lemmas are true, since their proofs have been checked and deemed correct by several mathematicians.

Research paper thumbnail of Occam's Razor and Computer-Program Testing

IEEE Transactions on Reliability, Aug 1, 1982

Occam's Razor and * The proof seems feasible for a class of programs which I Computer-Program... more Occam's Razor and * The proof seems feasible for a class of programs which I Computer-Program Testing have called O-Class ('O' for Occam's Razor). This class includes not only programs for addition and subtraction (which can be performed by finite-state machines) but also Peter Schorer for multiplication, division, and exponentiation. These latHewlett-Packard Co., Palo Alto ter three functions require arbitrarily large memory, hence cannot be performed by finite-state machines. Thus O-Class has a claim to being a useful class. Indeed

Research paper thumbnail of Is There a "Simple" Proof of Fermat's Last Theorem? Part (4)

concerned with ordered triples <a k , b k , c k >, where a, b, c, k are positive integers. In par... more concerned with ordered triples <a k , b k , c k >, where a, b, c, k are positive integers. In particular, we will be concerned with <x p , y p , z p >, where x p + y p = z p is an assumed minimum counterexample, and with all <x k , y k , z k >, where k  1. At times, for reasons that will become clear, we will also be concerned with ordered pairs, <x k + y k , z k >. (2) that, for a given u, as the modulus m increases, the location of u descends in the lines-andcircles model for each modulus. There exists a minimum m such that u < m. We say that u touches down at m. Clearly, u < m´ for all m´ > m. Informally, we say "once down, always down." Definition of "Appropriate Modulus" Fermat's Little Theorem states that if p is prime, then a  a p mod p. No restriction is placed on a-that is, it is not required that (a, p) = 1. On the other hand, Euler's generalization of Fermat's Little Theorem states that only under the conditions that (a, m) = 1 is it the case that a  a (m) + 1 mod m when m is composite. (The function m is Euler's totient function; its value is the number of positive integers less than m and relatively prime to m. If q is a prime, then q = q-1.) Throughout this section, therefore, when m is composite we will assume that this restriction is placed on any a, b, c-including x, y, z-that are involved in congruences mod m, and we will usually specify this by referring to m as an appropriate modulus. Definition of "Congruent Ordered Triples" Let <a k , b k , c k >, <a´ k´, b´ k´, c´ k´> be ordered triples, where a, b, c, a´, b´, c´, k, k´ are positive integers. Then if, for an appropriate modulus m, a k a´ k´, b k  b´ k´, and c k  c´ k´m od m, we say that the ordered triples are congruent mod m and that <a k , b k , c k > is congruent to <a´ k´, b´ k´, c´ k´> mod m. We will omit mod m when m is understood. For a triple <a k , b k , c k > there are two possibilities: a k + b k  c k mod m, or a k + b k not  c k mod m. In the first case, we say that the triple is a congruent triple, and in the second case we say that the triple is a non-congruent triple. It is important to understand that a finite or infinite set of congruent ordered triples (first sense) may contain ordered triples whose elements are congruent or non-congruent in the second sense. Definition of a Triple Being "Below" or "Lower Than" Than Another Triple Given two congruent triples, if each element of the first is less than the corresponding element of the second, we say that the first triple is below, or lower than, the second. Definition of "U(k, a, b, c)" Let k, a, b, c be positive integers. Then U(k, a, b, c) = a k + b k-c k. If x p + y p-z p is a minimum counterexample, we often abbrieviate U(k, x, y, z) to U k. Brief Description of Approach Type I Details on other Approaches are given in "Appendix A-Other Supporting Material for Approaches Based on the "Lines-and-Circles" Model of Congruence" on page 5. In this Approach, we try to show that the triples <x p , y p , z p > and <a k , b k , c k > give rise to a contradiction. We attempt to do this via two implementations: First Implementation: show that the contradiction arises between <a k , b k , c k > that are con

Research paper thumbnail of A Proof of Goldbach's Conjecture

Goldbach's Conjecture, which was announced in 1742, asserts that each even positive integer great... more Goldbach's Conjecture, which was announced in 1742, asserts that each even positive integer greater than or equal to 4 is the sum of two prime integers. Thus, e.g., 12 = 5 + 7. The Conjecture is still unproved.

Research paper thumbnail of The Structure of the 3x + 1 Function: An Introduction

This section contains statements of all results that we have obtained to date. A few of the resul... more This section contains statements of all results that we have obtained to date. A few of the results are already known in the literature, but are included for ease of reference. The reader is encouraged to use the“Table of Symbols and Terms” on page 103 to look up definitions of terms, and, of course, in any of our papers, to use the Search facility that is available with all .pdf files on a web site. Proofs that are not given in this paper are given in the papers, “The Structure of the 3x + 1 Function”, “Are We Near a Solution to the 3x + 1 Problem?” and “A Solution to the 3x + 1 Problem” on the web site www.occampress.com. The term “[so]” following a lemma number means that the statement and proof of the lemma will be found in the paper, “A Solution to the 3x + 1 Problem” on the web site www.occampress.com. The term “[ar]” following a lemma number means that the statement and proof of the lemma will be found in the paper, “Are We Near a Solution to the 3x + 1 Problem?” on the web s...

Research paper thumbnail of A Possible Exception to a Basic Principle of Special Relativity

Research paper thumbnail of Occam’s Razor and Program Proof by Test

It is well known that addition can be performed by a finite-state machine, but multiplication can... more It is well known that addition can be performed by a finite-state machine, but multiplication cannot, because of the need to store the intermediate sums, which for arbitrarily large x or y requires arbitrarily large memory. Therefore, in principle, it would seem that we can never hope to determine if a program "like" mult is correct merely by subjecting the program to a finite set of tests. Yet, at the same time, it seems hard to believe that at least certain types of error in mult could not be detected by testing. For example, suppose that the if clause were, "if x = 0 then 1 else...". This error would appear via a test of, e.g., mult(0, 3). The output would be 1 (incorrect) instead of 0 (correct). Or suppose that the recursive portion were "(y + 1) + mult(x-1, y)". This error would appear via a test of, e.g., mult(2, 1). The output would be 3 (incorrect) instead of 2 (correct). Example 2 Consider the following recursive program for the factorial function, f(n) = n! = n(n-1)(n-2)...1: [2] fact(n) == {if n = 1 then 1 else n fact(n-1)}. Such a program cannot be a finite-state machine because it must do multiplication. Therefore, in principle, it would seem that we can never hope to determine if a program "like" fact is correct merely by subjecting the program to a finite set of tests. Yet, as with mult, it seems hard to believe that at least certain types of error in fact could not be detected by testing, e.g., an error in the if clause such as "if n = 1 then 2" or an error in the recursive part such as "(n + 1)  fact(n-1)". Example 3 Suppose that someone has chosen two real numbers, x, y, i.e., u = 0.U 1 U 2 U 3 ..., v = 0.V 1 V 2 V 3 ..., where: U i , V i , i are decimal digits. This person will reveal the numbers to us one pair of digits at a time (up to some finite number of pairs), progressing from left to right. That is, he or she will reveal U 1 , V 1 , then U 2 , V 2 , etc. Following his or her revealing each pair of digits, we are to make one of three replies: (1) "u v." (2) "I am undecided whether u = v." (3) "u = v." Obviously, if, for some i, U i  V i , we will make reply (1). However, in general, we can never make reply (3) since, no matter how large the i such that U j = V j , 1  j  i, it is always possible that U i+1  V i+1. Suppose, however, before showing us any digits, the person tells us that u was generated by a Turing machine of x instructions and that v was generated by a Turing machine of y instructions. Furthermore, the person tells us (truthfully) that the Turing machines belong to a class of machines that are capable of expressing all computable functions, hence all computable numbers. Would there be a sufficiently large i such that, if U j = V j , 1  j  i, then we would be prepared to make reply (3)?

Research paper thumbnail of On “O1-Class” computer programs

Computers & Electrical Engineering, 1984

A definition is given of a class of computer programs, called the "O1-Class", in which the correc... more A definition is given of a class of computer programs, called the "O1-Class", in which the correctness of any program in the class relative to any other program in the class, over the countable infinity of all inputs, can be established by testing. The class is larger than that of finite-state machines. It is shown that a version of Occam's Razor is valid for O1-Class; i.e. if two programs in the class are being tested against a third program in the class, then, over a range of inputs related to the length of the three programs, the maximum number of tests required to detect an error in the program having fewer instructions is always less than or equal to the maximum number of tests required to detect an error in the program having the greater number of instructions.

Research paper thumbnail of Is There a 'Simple' Proof of Fermat's Last Theorem? (Part 1)

We present several approaches to a possible "simple" proof of Fermat's Last Theorem (FLT), which ... more We present several approaches to a possible "simple" proof of Fermat's Last Theorem (FLT), which states that for all n greater than 2, there do not exist x, y, z such that x n + y n = z n , where x, y, z, n, are positive integers. Until the mid-1990s, when a proof was given by Andrew Wiles, this had been the most famous unsolved problem in mathematics. But Wiles' proof was well over 100 pages long, and involved some of the most advanced mathematics of its time, and so the question lingers, "Is there a 'simple' proof of the Theorem?" Note 1: Without question, our three best Approaches to a simple proof of FLT are:

Research paper thumbnail of Occam's Razor and Program Proof by Test

Research paper thumbnail of Notes on Self-Representing, and Other, Information Structures

Motivation for These Notes These Notes have their foundations in three sources: first, and probab... more Motivation for These Notes These Notes have their foundations in three sources: first, and probably most important, Chaitin's papers an algorithmic information theory (see Bibliography); second, Mandelbrot's Fractals:

Research paper thumbnail of Structure the Use: Notes on a Method for Designing Computing System Environments

Research paper thumbnail of A Solution to the 3x + 1 Problem

Drafts by Peter Schorer

Research paper thumbnail of Solution to the Golbach’s Conjecture

Solution to the Golbach’s Conjecture

The solution here provided uses what call the Cantor-Schorer map inspired by Cantor's proof of th... more The solution here provided uses what call the Cantor-Schorer map inspired by Cantor's proof of the cardinality of the rational numbers and the matrix constructed by Peter Schorer in his "A Possible Proof of Goldbach's Conjecture" Discussion Let's consider the mapping between prime and even numbers, shown in the following figure. This is an infinite matrix where the first row and the first column enumerate all the prime numbers. Using Schorer's construction, every cell contains the sum of the corresponding row and column. The line connects all the cells in the matrix. This means that it covers all the prime pairs. Observe that the inner cells completely cover the set of even numbers from 6 (to be proved). But by construction, these numbers can only be the sum of some two primes.

Research paper thumbnail of A Very Short Proof of Fermat's Last Theorem

Research paper thumbnail of Is a solution to the 3x + 1 problem in sight?

SIGACT news, Dec 1, 1997

Dec. 14, 2024 "Very often in mathematics the crucial problem is to recognize and to discover what... more Dec. 14, 2024 "Very often in mathematics the crucial problem is to recognize and to discover what are the relevant concepts; once this is accomplished the job may be more than half done." 1 "One of the greatest contributions a mathematician can make is to spot something so simple and powerful that everybody else has missed it." 2 Note 1 : Our proofs of the 3x + 1 Conjecture begin under "Proofs of the 3x + 1 Conjecture " on page 3. (A proof of the Conjecture solves the 3x + 1 Problem.) The reader should begin reading on page 2. Note 2: We will pay 75,000tothenumbertheoristwhohelpsusprepareaversionofthispaperthatresultsinourwlnningthenearly75,000 to the number theorist who helps us prepare a version of this paper that results in our wlnning the nearly 75,000tothenumbertheoristwhohelpsusprepareaversionofthispaperthatresultsinourwlnningthenearly1 million prize for a solution to the 3x + 1 problem that is being offered by a Japanese organization. See details in Appendix D at the end of this paper.

Research paper thumbnail of A Solution to the 3x + 1 Problem

The 3x + 1 Problem asks if repeated iterations of the function C(x) = (3x + 1)/(2a) always termin... more The 3x + 1 Problem asks if repeated iterations of the function C(x) = (3x + 1)/(2a) always terminate in 1 Here x is an odd, positive integer, and a = ord2(3x + 1), the largest positive integer such that the denominator divides the numerator. The conjecture that the function always eventually terminates in 1 is the 3x + 1 Conjecture. An odd, positive integer that maps to 1 is called a non-counterexample; an odd, positive integer that doesn’t map to 1 is called a counterexample (to the Conjecture). Our first proof (given under:“First Proof of the 3x + 1 Conjecture” on page 11) is based on a structure called tuple-sets that represents the 3x + 1 function in the “forward” direction. In our proof, we show that the 35-level elements of all 35-level tuples in all 35-level tuple-sets are the same, regardless if counterexamples to the Conjecture exist or not1. From this fact, a simple inductive argument allows us to conclude that all tuple-sets are the same, whether counterexamples exist or not, and hence that counterexamples do not exist. Our second proof (given in “Second Proof of the 3x + 1 Conjecture” on page 12), like the first, is based on tuple-sets. In this proof, we define anchor, which is the i-level element of the first i-level tuple in an i-level tuple-set. We then show that there is one and only one set of anchors for all i, regardless if counterexamples exist or not. We then show that this implies that there is one and only one set of infinite tuples, regardless if counterexamples exist or not, and from this we deduce that, if counterexamples exist, then some infinite tuples must be both counterexample and non-counterexample tuples, which is absurd, hence counterexamples do not exist and the Conjecture is true. Our third proof (given in “Third Proof of the 3x + 1 Conjecture” on page 21) has three versions. The first is based on a structure called the 1-tree. This tree is a y-tree, where the root y is a range element of the 3x + 1 function; y-trees represent the 3x + 1 function in the “inverse” direction. The second version is based on the remarkable fact that each finite sequence of iterations of the 3x + 1 function can in principle be traced on a certain spiral diagram. The third version is also based on the 1-tree. As far as we have been able to determine, our approaches to a solution of the Problem are original. 1. A phrase of the form “q regardless if p” is equivalent to “(if p then q) and (if not-p then q)”. It is meaningful and in fact true as long as q is true, which it always is in this paper. Instances of the phrase occur in everyday speech, for example, “Fermat’s Last Theorem is true regardless if the Riemann Conjecture is true”.,

Research paper thumbnail of Structure the Use: Notes on a Method for Designing Computing System Environments

IEEE Computer, Dec 1, 1981

We may be able to design a better computer system if we think of it as an entity inseparable from... more We may be able to design a better computer system if we think of it as an entity inseparable from all of the training, documentation, and folklore that help make it work.

Research paper thumbnail of A Discussion of Several Possible Strategies

Readers can safely assume, initially, that all referenced lemmas are true, since their proofs hav... more Readers can safely assume, initially, that all referenced lemmas are true, since their proofs have been checked and deemed correct by several mathematicians.

Research paper thumbnail of Occam's Razor and Computer-Program Testing

IEEE Transactions on Reliability, Aug 1, 1982

Occam's Razor and * The proof seems feasible for a class of programs which I Computer-Program... more Occam's Razor and * The proof seems feasible for a class of programs which I Computer-Program Testing have called O-Class ('O' for Occam's Razor). This class includes not only programs for addition and subtraction (which can be performed by finite-state machines) but also Peter Schorer for multiplication, division, and exponentiation. These latHewlett-Packard Co., Palo Alto ter three functions require arbitrarily large memory, hence cannot be performed by finite-state machines. Thus O-Class has a claim to being a useful class. Indeed

Research paper thumbnail of Is There a "Simple" Proof of Fermat's Last Theorem? Part (4)

concerned with ordered triples <a k , b k , c k >, where a, b, c, k are positive integers. In par... more concerned with ordered triples <a k , b k , c k >, where a, b, c, k are positive integers. In particular, we will be concerned with <x p , y p , z p >, where x p + y p = z p is an assumed minimum counterexample, and with all <x k , y k , z k >, where k  1. At times, for reasons that will become clear, we will also be concerned with ordered pairs, <x k + y k , z k >. (2) that, for a given u, as the modulus m increases, the location of u descends in the lines-andcircles model for each modulus. There exists a minimum m such that u < m. We say that u touches down at m. Clearly, u < m´ for all m´ > m. Informally, we say "once down, always down." Definition of "Appropriate Modulus" Fermat's Little Theorem states that if p is prime, then a  a p mod p. No restriction is placed on a-that is, it is not required that (a, p) = 1. On the other hand, Euler's generalization of Fermat's Little Theorem states that only under the conditions that (a, m) = 1 is it the case that a  a (m) + 1 mod m when m is composite. (The function m is Euler's totient function; its value is the number of positive integers less than m and relatively prime to m. If q is a prime, then q = q-1.) Throughout this section, therefore, when m is composite we will assume that this restriction is placed on any a, b, c-including x, y, z-that are involved in congruences mod m, and we will usually specify this by referring to m as an appropriate modulus. Definition of "Congruent Ordered Triples" Let <a k , b k , c k >, <a´ k´, b´ k´, c´ k´> be ordered triples, where a, b, c, a´, b´, c´, k, k´ are positive integers. Then if, for an appropriate modulus m, a k a´ k´, b k  b´ k´, and c k  c´ k´m od m, we say that the ordered triples are congruent mod m and that <a k , b k , c k > is congruent to <a´ k´, b´ k´, c´ k´> mod m. We will omit mod m when m is understood. For a triple <a k , b k , c k > there are two possibilities: a k + b k  c k mod m, or a k + b k not  c k mod m. In the first case, we say that the triple is a congruent triple, and in the second case we say that the triple is a non-congruent triple. It is important to understand that a finite or infinite set of congruent ordered triples (first sense) may contain ordered triples whose elements are congruent or non-congruent in the second sense. Definition of a Triple Being "Below" or "Lower Than" Than Another Triple Given two congruent triples, if each element of the first is less than the corresponding element of the second, we say that the first triple is below, or lower than, the second. Definition of "U(k, a, b, c)" Let k, a, b, c be positive integers. Then U(k, a, b, c) = a k + b k-c k. If x p + y p-z p is a minimum counterexample, we often abbrieviate U(k, x, y, z) to U k. Brief Description of Approach Type I Details on other Approaches are given in "Appendix A-Other Supporting Material for Approaches Based on the "Lines-and-Circles" Model of Congruence" on page 5. In this Approach, we try to show that the triples <x p , y p , z p > and <a k , b k , c k > give rise to a contradiction. We attempt to do this via two implementations: First Implementation: show that the contradiction arises between <a k , b k , c k > that are con

Research paper thumbnail of A Proof of Goldbach's Conjecture

Goldbach's Conjecture, which was announced in 1742, asserts that each even positive integer great... more Goldbach's Conjecture, which was announced in 1742, asserts that each even positive integer greater than or equal to 4 is the sum of two prime integers. Thus, e.g., 12 = 5 + 7. The Conjecture is still unproved.

Research paper thumbnail of The Structure of the 3x + 1 Function: An Introduction

This section contains statements of all results that we have obtained to date. A few of the resul... more This section contains statements of all results that we have obtained to date. A few of the results are already known in the literature, but are included for ease of reference. The reader is encouraged to use the“Table of Symbols and Terms” on page 103 to look up definitions of terms, and, of course, in any of our papers, to use the Search facility that is available with all .pdf files on a web site. Proofs that are not given in this paper are given in the papers, “The Structure of the 3x + 1 Function”, “Are We Near a Solution to the 3x + 1 Problem?” and “A Solution to the 3x + 1 Problem” on the web site www.occampress.com. The term “[so]” following a lemma number means that the statement and proof of the lemma will be found in the paper, “A Solution to the 3x + 1 Problem” on the web site www.occampress.com. The term “[ar]” following a lemma number means that the statement and proof of the lemma will be found in the paper, “Are We Near a Solution to the 3x + 1 Problem?” on the web s...

Research paper thumbnail of A Possible Exception to a Basic Principle of Special Relativity

Research paper thumbnail of Occam’s Razor and Program Proof by Test

It is well known that addition can be performed by a finite-state machine, but multiplication can... more It is well known that addition can be performed by a finite-state machine, but multiplication cannot, because of the need to store the intermediate sums, which for arbitrarily large x or y requires arbitrarily large memory. Therefore, in principle, it would seem that we can never hope to determine if a program "like" mult is correct merely by subjecting the program to a finite set of tests. Yet, at the same time, it seems hard to believe that at least certain types of error in mult could not be detected by testing. For example, suppose that the if clause were, "if x = 0 then 1 else...". This error would appear via a test of, e.g., mult(0, 3). The output would be 1 (incorrect) instead of 0 (correct). Or suppose that the recursive portion were "(y + 1) + mult(x-1, y)". This error would appear via a test of, e.g., mult(2, 1). The output would be 3 (incorrect) instead of 2 (correct). Example 2 Consider the following recursive program for the factorial function, f(n) = n! = n(n-1)(n-2)...1: [2] fact(n) == {if n = 1 then 1 else n fact(n-1)}. Such a program cannot be a finite-state machine because it must do multiplication. Therefore, in principle, it would seem that we can never hope to determine if a program "like" fact is correct merely by subjecting the program to a finite set of tests. Yet, as with mult, it seems hard to believe that at least certain types of error in fact could not be detected by testing, e.g., an error in the if clause such as "if n = 1 then 2" or an error in the recursive part such as "(n + 1)  fact(n-1)". Example 3 Suppose that someone has chosen two real numbers, x, y, i.e., u = 0.U 1 U 2 U 3 ..., v = 0.V 1 V 2 V 3 ..., where: U i , V i , i are decimal digits. This person will reveal the numbers to us one pair of digits at a time (up to some finite number of pairs), progressing from left to right. That is, he or she will reveal U 1 , V 1 , then U 2 , V 2 , etc. Following his or her revealing each pair of digits, we are to make one of three replies: (1) "u v." (2) "I am undecided whether u = v." (3) "u = v." Obviously, if, for some i, U i  V i , we will make reply (1). However, in general, we can never make reply (3) since, no matter how large the i such that U j = V j , 1  j  i, it is always possible that U i+1  V i+1. Suppose, however, before showing us any digits, the person tells us that u was generated by a Turing machine of x instructions and that v was generated by a Turing machine of y instructions. Furthermore, the person tells us (truthfully) that the Turing machines belong to a class of machines that are capable of expressing all computable functions, hence all computable numbers. Would there be a sufficiently large i such that, if U j = V j , 1  j  i, then we would be prepared to make reply (3)?

Research paper thumbnail of On “O1-Class” computer programs

Computers & Electrical Engineering, 1984

A definition is given of a class of computer programs, called the "O1-Class", in which the correc... more A definition is given of a class of computer programs, called the "O1-Class", in which the correctness of any program in the class relative to any other program in the class, over the countable infinity of all inputs, can be established by testing. The class is larger than that of finite-state machines. It is shown that a version of Occam's Razor is valid for O1-Class; i.e. if two programs in the class are being tested against a third program in the class, then, over a range of inputs related to the length of the three programs, the maximum number of tests required to detect an error in the program having fewer instructions is always less than or equal to the maximum number of tests required to detect an error in the program having the greater number of instructions.

Research paper thumbnail of Is There a 'Simple' Proof of Fermat's Last Theorem? (Part 1)

We present several approaches to a possible "simple" proof of Fermat's Last Theorem (FLT), which ... more We present several approaches to a possible "simple" proof of Fermat's Last Theorem (FLT), which states that for all n greater than 2, there do not exist x, y, z such that x n + y n = z n , where x, y, z, n, are positive integers. Until the mid-1990s, when a proof was given by Andrew Wiles, this had been the most famous unsolved problem in mathematics. But Wiles' proof was well over 100 pages long, and involved some of the most advanced mathematics of its time, and so the question lingers, "Is there a 'simple' proof of the Theorem?" Note 1: Without question, our three best Approaches to a simple proof of FLT are:

Research paper thumbnail of Occam's Razor and Program Proof by Test

Research paper thumbnail of Notes on Self-Representing, and Other, Information Structures

Motivation for These Notes These Notes have their foundations in three sources: first, and probab... more Motivation for These Notes These Notes have their foundations in three sources: first, and probably most important, Chaitin's papers an algorithmic information theory (see Bibliography); second, Mandelbrot's Fractals:

Research paper thumbnail of Structure the Use: Notes on a Method for Designing Computing System Environments

Research paper thumbnail of A Solution to the 3x + 1 Problem

Research paper thumbnail of Solution to the Golbach’s Conjecture

Solution to the Golbach’s Conjecture

The solution here provided uses what call the Cantor-Schorer map inspired by Cantor's proof of th... more The solution here provided uses what call the Cantor-Schorer map inspired by Cantor's proof of the cardinality of the rational numbers and the matrix constructed by Peter Schorer in his "A Possible Proof of Goldbach's Conjecture" Discussion Let's consider the mapping between prime and even numbers, shown in the following figure. This is an infinite matrix where the first row and the first column enumerate all the prime numbers. Using Schorer's construction, every cell contains the sum of the corresponding row and column. The line connects all the cells in the matrix. This means that it covers all the prime pairs. Observe that the inner cells completely cover the set of even numbers from 6 (to be proved). But by construction, these numbers can only be the sum of some two primes.