The Riemann hypothesis is proved in Number, space
and logic, volume III chapter V, under informal review.
The Superexponential algebra section has a reviewed
chapter on novanions, used in the Physics section.
We replace Galois theory using dependent roots, ring
automorphisms and varieties. The quintic is not solvable
by 'killing central terms' but it is solvable - in radicals by
'nondescending comparison', and by approximation methods.
We prove that consistent problems are finitely or infinitely
decidable, so the only undecidable problems are inconsistent,
and show finitely stated problems can be determined finitely.
eBook 2017 Superexponential algebra I, II & III
eBook 2018 Number, space and logic I, II & III (unfinished)
eBook 2014 Innovation in mathematics print
eBook 2009 Elementary methods in number theory
This mathematics section describes the research programme. A brief history is given here and the research schedule.
An invitation for you to get involved in these research areas.
I have a number of papers and ideas, and the most significant are posted here. Would you like to develop some of these ideas further, or offer criticism of any paper? I would welcome comments not only from professional mathematicians. If you wish to email me, the email address is at the top of the website.
1. Research Programme Synopsis is a brief technical sketch of the generalisation programme in mathematics.
2. Research Programme covers all of my research topics in mathematics.
Return to Table of Contents.
You can comment on this eBook, Superexponential Algebra.
1A. Foreword The Foreword, Table of Contents and Mathematical terms of Superexponential Algebra Volume I.
1B. Prologue Why is mathematics there? and What is mathematics? of Superexponential Algebra.
1C. Intricate Numbers Chapter I introduces the intricate matrix representation.
1D. Hyperintricate Numbers Chapter II introduces the hyperintricate matrix representation.
1E. Associative Division Algebras Chapter III. This chapter contains a proof that the only standard associative division algebras are the reals, complex numbers and quaternions. In the case where there is more than one axis with square 1, a restricted set of singularities is present. We also give a proof of Wedderburn's little theorem.
1F. Nonassociative Algebras Chapter IV. We introduce nonassociative operations derived from associative matrix multiplication. This extends the hyperintricate methodology. We also discuss Lie and Kac-Moody algebras.
1G. Novanions Chapter V. This chapter contains a proof that there exist nonassociative novanion algebras of dimension higher than the octonions. Novanion algebras are division algebras, except when the scalar part is zero, when the product of two nonzero novanions can be zero. We introduce explicit models, including the ten dimensional 10-novanions. For applications, see the Physics section.
1H. Fermat's Little Theorem for Matrices Chapter VI uses the hyperintricate representation of matrices to generalise Fermat's little theorem (not identical to its non-matrix cousin) and the Euler totient formula.
2A. Foreword The Foreword, Table of Contents and Mathematical terms of Superexponential Algebra Volume II.
2B. Ladder and Complex Algebra Chapter VII. This chapter discusses the incompatibility between the continuum hypothesis and the countability of the rational numbers, and introduces winding numbers to prove the fundamental theorem of algebra.
2C. Polynomials with Complex Roots Chapter VIII. We investigate the sextic Bring-Jerrard polynomial. For independent roots this chapter assumes Galois theory holds. For dependent roots we give a description of polynomial entities and show they are solvable. These entities always contain a general polynomial. We give a general theory of dependent roots.
2D. Polynomials with Matrix Roots Chapter IX. This chapter deals with matrix variables. Using the Cayley-Hamilton theorem and companion matrices, we show matrix solutions for complex polynomials of any degree can be found. For independent roots we investigate matrix solutions of polynomial equations with intricate coefficients up to the quartic. The fundamental theorem of algebra fails for matrix polynomials, and solution conditions can be more severe than the ordinary case.
2E. Automorphisms and Linear Maps of Polynomial Equations Chapter X. This chapter shows that ring automorphisms for polynomials are commutative, unlike inner group automorphisms, so Galois theory fails.
2F. Solvability of Complex Varieties Chapter XI. This chapter shows polynomial equations are equivalent to varieties in two variables. The end result of Galois theory - no solutions of general complex polynomial equations of degree greater than 4 by radicals - follows in the case of 'killing central terms' of polynomial equations, and other descending methods are equivalent to this. However, we are able to express a cubic polynomial entirely in terms of square roots, which is impossible by Galois theory, but we fail here to solve the sextic equation in radicals by 'nondescending comparison' methods. QR matrix approximation methods are also discussed.
2G. Polynomial Rings and Ideals Chapter XII. We introduce Hilbert's basis theorem, the Nullstellensatz and Groebner bases.
2H. Probability Sheaves Chapter XIII. The contents of my work here go back 37 years.
3A. Foreword The Foreword, Table of Contents and Mathematical terms of Superexponential Algebra Volume III.
3B. Algorithms and Consistency Chapter XIV. This chapter looks at nonstandard interpretations of the continuum hypothesis and discusses decidability and consistency with nonstandard outcomes in our developments of set and number theory.
3C. Exponential Algebra Chapter XV is an introductory chapter on hyperintricate exponentiation.
3D. The Dw Exponential Algebras Chapter XVI modifies complex exponentiation and extends it to the hyperintricate proposal D1, which addresses in detail a problem puzzling me for forty years. Although roots do branch, the algebra suggested does not give branched real values, other than plus or minus a real root, unlike the usual algebra. Exponentiation is the first non-inverse operation which is non-associative and it is not in some other ways like a group.
3E. Superexponentiation Chapter XVII. These superexponentiation operations generalise eaeb = ea + b. Mappings discussed here are not associative.
3F. Appendices, Answers to Exercises, References and Index Chapter XVIII. The concluding part of the eBook.
The Mathematical archive accesses historical material that went into the creation of Superexponential algebra.
Return to Table of Contents.
Number, space and logic is half complete, containing ten months work. It will extend
Fred Diamond and Jerry Shurman, A first course in modular forms, Springer, 2005, which is strongly recommended,
John H. Conway and Derek Smith, On Quaternions and Octonions, A. K. Peters, 2003,
Hershel Farkas and Irwin Kra, Riemann Surfaces (2nd edn.), Springer, 1992 and
M.E. Szabo (ed.) The collected papers of Gerhard Gentzen, North-Holland, 1969.
There will be an exposition and extension of homolgy, cohomology and homotopy theory within it.
The curent contents, which may contain conceptual and other errors, are
VOLUME I: Superstructures
1A. Foreword Foreword and table of contents.
1B. The Meaning of the Finite and the Infinite Chapter 1. This chapter introduces the Peano axioms for the natural numbers, and extends this system to transfinite natural numbers, which we call transnatural numbers. Some basic properties of the transnatural numbers are developed, we deconstruct the uncountable continuum hypothesis for the set of subsets of the natural numbers, and we introduce an algebra of transinfinite ordinals, called ladder algebra.
1C. The Meaning of Branched Spaces Chapter 2. We introduce the idea of branched space topology which acts as a model for our extended idea of the Euler characteristic, as a general polynomial with integer coefficients.
1D. The Meaning of Superstructures Chapter 3. This chapter introduces a general algebra for superexponentiation in an extended form from that given in the eBook Superexponential algebra.
1E. The Novanion Simple Group Construction Chapter 4 is not yet complete. This chapter links the classification of simple groups with novanions. It says that we obtain Lie algebras beyond those obtained from the octonions, with more elements than E8. 10- and 26-novanions have both fermionic and bosonic representations. We obtain some simple groups with similar but different size to the monster, and an infinite sequence of such simple groups.
1F. Polynomial Comparison Chapter 5. This chapter surveys the results of Superexponential algebra volume II which demonstrates that the Galois solvability model is incorrect for comparison methods which avoid killing central terms, shows by general methods that solutions of polynomial equations of arbitrary degree can always be found in principle, provides a geometric realisation of the cubic, and looks in detail at the solvability of quintic polynomials via comparison with polynomials of the sixth and eighth degrees. Our initial attempts here fail, but we review the solvability of the quintic as embedded in a quartic variety in squared variables, which process solves this problem. We are able to show that the quintic gives a solution of the sextic. This is part of a more general theorem that a solvable polynomial of odd degree n gives rise to a solvable polynomial of degree n + 1. The septic is demonstrated to be solvable by radicals in a similar way to the quintic, by embedding in a sextic variety in squared variables. Thus equations up to the octic can be solved, and this process can be iterated further, as part of a general theorem that a solvable polynomial of even degree n gives rise to a solvable polynomial of degree n + 1 .
1G. Novanion Varieties Chapter 6 is not ready. This chapter contains a description of the theory of elliptic curves.
1H. Modular Forms Chapter 7 is not ready. This chapter includes work on modular forms, which is used in the modularity theorem.
1I. Hyperintricate Analysis Chapter 8 is half ready. This chapter includes work on the hyperintricate Cauchy-Riemann equations and the Cauchy integral formula.
1J. Superanalysis Chapter 9 is not ready. This develops superexponential analogues of the previous chapter.
VOLUME II: Trees and Amalgams
2A. Foreword Foreword and table of contents.
2B. Trees and Amalgams Chapter 1 is not ready. This chapter defines trees and amalgams.
2C. Branched Spaces and Explosions Chapter 2 is not ready. This chapter describes branching which is a general feature of superexponential algebras. The branching can be infinite in a set, called an explosion. If the algebraic structure is removed, the topology remains. This chapter includes work on the Riemann-Roch theorem, the Gauss-Bonnet theorem and stereographic projection.
2D. Surgery Chapter 3 is not ready. This chapter introduces the idea of obtaining from the branched Euler characteristic in multiplicative form, a series of operations called surgery to enhance the polynomial to a desired additive form. This algebraic series of operations has a geometric realisation.
2E. Sequent Calculus and Colour Logic Chapter 4 is not ready. This chapter looks at multivalued logics, called colour logics, develops Gentzen's sequent calculus and extends the discussion of sequents to modal and colour logic.
2F. Homology Chapter 5 is not ready. This chapter describes our replacement of homology in the superexponential context.
2G. Cohomology Chapter 6 is not ready. This chapter describes etale and motivic cohomology in the superexponential context.
2H. Homotopy Chapter 7 is not ready. This chapter describes homotopy, a theory of paths, in the superexponential context.
VOLUME III: The Finite and the Infinite
3A. Foreword Foreword and table of contents.
3B. Number Theory Chapter 1 is half constructed. This chapter looks at ladder algebra and its transcendental extensions. It has a section on Gauss, Ramanujan and Kloosterman sums.
3C. Class Field Theory Chapter 2 is not ready. This chapter looks at class field theory.
3D. The Consistency of Analysis Chapter 3 is not ready. This chapter develops a Gentzen-type proof of the consistency of analysis from our theory of trees and our transcendental results. This theory is extended to cover colour logics, an implication of this being an extension to intuitionalistic logics.
3E. Fermat's Last Theorem Chapter 4. This chapter proves Fermat's Last Theorem using elementary methods. We also give a more sophisticated treatment using the modularity theorem.
3F. Zeta Functions Chapter 5. This chapter proves the classical theory of the Riemann hypothesis using invariance under imaginary Dw exponential algebras and the general Riemann hypothesis by extending w to include a rational component.
3G. Local Zeta Functions Chapter 6 is not ready. This chapter looks at the Weil conjectures. Implications for the function field case are given, where transl-adic and other results can be applied.
3H. The Goldbach Conjecture Chapter 7 is not ready. This chapter proves the weak Goldbach theorem using the work of Harald Helfgott. The general Riemann hypothesis implies a new and shorter technique for proving the weak Goldbach conjecture.
3I. References and Index Not ready.
The topology work in the archive was to become part of Number, space and logic - it describes branched spaces and their Euler characteristics. I would like to thank James Hirschfeld for getting a research student to look at the homology and cohomology. The following is this archive material.
The Mathematical archive accesses historical material that was to go into the creation of Number, space and logic.
Return to Table of Contents.
Foreword and Table of Contents
1. Creative Mathematics is a primer with suggestions on producing creative mathematics.
2. Discussion on Ladder Numbers shows for the general reader and undergraduate mathematician the inconsistency of the real numbers and aspects of ladder numbers and zero algebras, by introducing an algebra for infinitesimals compatible with nonstandard analysis, and for infinities. A definition of relative countability is used to challenge Cantor's diagonal argument, so using transfinite induction we show the inconsistency of the real number system, as currently axiomatised, and provide an alternative. The zero algebras equate 0/0 to a number, or a set of numbers.
3. Polynomial Equations I. Duplicate Roots. A solution by radicals of the zeros of the sextic polynomial equation with duplicate roots has been obtained. This does not contradict the non-existence of a Jordan-Hoelder series
S5 --> S4 --> S3 --> S2 --> 1.
It has been obtained by a formal 'non inertial' differential condition. We also describe an assumed method of Grothendieck, which uses multifunctions and in the case of the sextic corresponds to the series
S6 --> S3 --> S2 --> 1.
4. Polynomial Equations II. Transcendental Solutions is a solution of the zeros of polynomial equations with complex coefficients of any large degree by transcendental methods. This is not in violation of Galois theory.
5. Intricate and Hyperintricate Numbers I is an introduction to the essentials of the subject. These numbers are generalisations of complex numbers, and are general representations of 2n X 2n matrices.
6. Intricate and Hyperintricate Numbers II is a more advanced continuation of Part I.
7. Totient Reciprocity. There are said to be 200 proofs of quadratic reciprocity. My original comment was: this makes it the 201st! - but it is similar to a proof by Kronecker, and the work generalises this theorem.
8. Quadratic Residues Power Point Presentation Power Point presentation of Parts I and II of the Quadratic Residues papers, investigating by elementary methods the case that obtains for p prime = 4k - 1 that there are more quadratic residues in the interval [1, 2k - 1] than in [2k, 4k - 2].
8A. Quadratic Residues I is the first stage of a solution of a problem unproved by elementary methods in 2004, that for p prime = 4k - 1 there are more quadratic residues in the interval [1, 2k - 1] than in [2k, 4k - 2]. This is called the total disparity. A formula is proved.
8B. Quadratic Residues II We describe row and trajectory regions containing parabolas.
8C. Quadratic Residues III We refer and relate the disparity for prime p = 4k - 1 to transcendental methods and the 'tenth discriminant' problem. We continue the description of regions with parabolas. We discuss the 'shifted' disparity for primes of the form q = 4k + 1, and discuss the corresponding cases for numbers 4k and 4k + 2.
9. Eisenstein on Reciprocity Theorems Eisenstein's 'Applications of algebra to transcendental arithmetic', which I found sufficiently insightful to translate into English.
References and Index
The Mathematical archive accesses historical material that went into the creation of Innovation in mathematics.
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Other early papers
which contains early introductory and simpler papers for a general readership.
1. Fermat Numbers and Two Prime Number Theorems relates the idea of Fermat numbers to two prime number theorems.
2. Vector Calculus, a note to a colleague, describes mathematical ideas on vector calculus.
3. Partitions is a little note on partitions, originally an email, prompted by Paul Hammond.
The eBook Elementary methods in number theory
Foreword and Table of Contents
1. Chapter I is about exponential powers.
2. Chapter II on prime numbers, factorisation and divisibility.
3. Chapter III on differences and sums of pth and different powers mod 4.
4. Chapter IV is on Quadronacci numbers, a generalisation of Fibonacci numbers.
5. Chapter V contains work on Fermat's last theorem by elementary methods.
6. Chapter VI is on Beal's conjecture, an extension of Fermat's last theorem.
References and Index
The Mathematical archive accesses historical material that went into the creation of Elementary methods in number theory.
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