## Welcome from Wallace Adams, Jim Adams, Doly Garcia and Jim Hamilton

## Please practice saying what you think. Feel free to disagree with what I say and prove a contradiction!

The Riemann hypothesis is proved in Number, space and logic (NSL), volume II chapter 6,

under informal review, using Dw exponential algebra developed by David Bohm and me.NSL volume I chapter 4, on adonions which are division algebras and novanions (these are all called

zargonions), is used in the Physics section. Adonions beyond the octonions exist but are supposed not to.Games of the form (+, 0, -) exist throughout social systems, mathematics and physics. Other ordered

game triples are (cooperative, control or competitive, uncooperative), (ethics, Kampf wall, murder),

in solving polynomials (greater degree comparison, stable degree module, lesser degree polynomial wheel)

and in physics (J^{2}= -1 global interaction zargonion, J^{2}= 0 tribble, J^{2}= 1 quantum tharl algebra) aligned by

'overwhelming'. So with the Kampf wall down, murder with death overwhelms left ethics with a currency of need.

For applications in engineering*conceptors*are meaning processors rather than syntax processors (computers).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 'polynomial wheels', and using approximation methods.**I have finally after 54 years refuted Galois solvability theory by proving the quintic is solvable.**

This has been my aim since the age of 15. The solution technique is simple but unusual:

(x^{3}+ Ax^{2}+ Bx + C)^{2}- (x^{3}+ Dx^{2}+ Ex + F)^{2}= 0, constraint A - D = C - F = k

and compare with the quintic in Bring-Jerrard form x^{5}+ Tx + U = 0. Find k, etc.

Most decisively of all, as ought to be evident, it generalises to all polynomial equations.

Every polynomial equation is solvable in radicals, and therefore every consistent algorithm.

The most alarming aspect is the control wall of the theory of modules, and thus homological algebra.

The theory is almost not there - a black hole. It is like the burning of the Library of Alexandria.We prove that consistent problems are finitely or infinitely decidable, so the only undecidable

problems are inconsistent. We show finitely stated problems can be determined finitely.**As our class had proved at my age of 16, the paradox of the Cretan is no paradox (it ignores brackets).**

We decided, quite correctly, that no academic publication could accept a half-page disproof

from a class of 16 year olds equivalent to refuting Godel's Incompleteness Theorem.

Anyone allowing that would be thought an ignoramus and prevented from teaching.

So Godel's Incompleteness Theorem is incorrect. It was widely felt Godel was nuts. Well, yes.

Godel gave Einstein a birthday present with general relativity for a galaxy with a time loop.

He died from starvation when his wife went in hospital. All his results are wrong, except the trivial.

For the Incompleteness Theorem, he neglected in his 290 page proof

that general recursive functions do not allow 1 = 0.

Gentzen, who proved the contrary, accepted the Godel peer review.

Gentzen also died of starvation, as a German in a Czechoslovak prison after the War.

He had been invited to Princeton by Herman Weyl. A reactionary mathematician I know who will not read my work

calls Gentzen nuts too, but he has never read his work. It is translated in Kleen's Introduction to Metamathematics

and intensively but badly described there in the last, advanced, section of the book.

In fact Gentzen is a bit basic. He uses logical deduction in sorts of Aristotelian syllogisms, for instance.

This is more general than Boolean algeba, which is a subcase. He is ignored.It would be productive to say: I am a mathematical

*insight*theorist

I can look through great wodges of data and get insight into what they mean.

This Insight theory is in fact exact; it back-propogates to axioms, deductions and theorems.

It treats these consistent algebras as variables in algebra itself.

That is why I am a great mathematician. I work with the*meaning*of results

rather than mathematical gumph. It works. It is a cognitive disability, and very much

it is the*best way*for all mathematical research. We**must**use insight.To speak the Truth to others

We have first to find it in Ourselves.

This is Very Difficult. It is Very difficult Indeed.The truth needs to be stated that it is not dependent on any social system

but is located in the Truth itself. This is the error of peer review.

Until this is expurgated from all mathematical research, mathematics can make no progress.

It is with deep regret that I inform you, despite at least 54 years of intensive research

revealing massive and deeply hidden confusions in the entirety so far as I can ascertain

of the mathematics of the twentieth centrury, the reluctant and astonishing result in this regard.

Including I thought the exception, which is a huge edifice of considerable mathematics of the

proof of the Weil conjectures, which ought to deal with the binomial theorem only, but for which

Galois solvabiity turns out to be deeply embedded as the theory of groups, mathematics is wrong.The best discoveries are of the self.

It is not the beauty of mistakes that we can improve them.

Nevertheless it appears minor tweaking

Can reveal New Profundities in Violation of Current results.Approximately, mathematics is 55% wrong, physics is 85% wrong and everything else is worse.

The problem with sociology is there is no mathematics developed to express how wrong it is,

apart from the puzzling Theodor Adorno's 'Against Epistemology' which it took me 38 years to

realise is an elaborate joke!This may be thought funny and distressing. For me it is so utterly distressing as to be beyond belief.

Scientists like fudge control. The documents of Mathematics must be accepted. My interest is elsewhere.

Having accepted that Galois Representation theory is correct, and bears no relation with Galois Solvability whatsoever,

intelligent scientists excluded from the system particularly for incompetence are needed to look at twentieth century mathematics.

This website has been reduced. More advanced material will be contained in the companion website www.theloveoftruth.ie, but is not yet set up.

Heuristic principle: if a result is technically erudite, needs extremely long reasoning and is counterintuitive, it is wrong.

ISBN 9780387979939.

*The Unmentionable **How to ascend University dogmas to purer dogmas nearer the truth***e****Book 2020 Number, space and logic I, II, III, IV & V (unfinished)**

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**Control Order**

*This is a Control Order from Daleks in the BBC.*

Publish the following equations in BBC comedy broadcasts

otherwise the consequences will be unmentionable

1 + 1 = 2, 2 + 2 = 5.

The 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. The most significant are posted here. Would you like to develop some of them further or offer criticism of any paper? I welcome comments not only

from professional mathematicians. 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 in mathematics.

I had thought that the truth of an idea in mathematics would eventually lead to its acceptance. From my investigation of social systems, I believe if a set of ideas deviates from a central

paradigm, it can never be accepted by an authoritarian social structure. Since the social structure of mathematics in all universities today of which I have knowledge are authoritarian, it

follows that the ideas given here will never be accepted by the current system. If we come up with the question of how much Orthodox Judaism has changed in the past 1000 years, we

would certainly say a lot, but I do not think it has upturned central paradigms. These are completely unacceptable, whatever their relation to truth. The same is the case in science today.

It is dogma, and resistant to changes in thinking which act against the social structure maintaining this dogma. The only recourse is to abandon this system and build a new one based on

social structures which are inherently adaptive. These new social systems cannot be accommodated in any authoritarian structure.

It may be objected that freedom of thought was allowed under restricted circumstances in authoritarian systems, for instance the Manhatten project in the US and mathematics research

in the Stalinist Soviet Union. Such exceptions are absent today. Intensification of the capitalist system has led to the replacement of egalitarian university systems with heirarchical ones

with power-centric Vice Chancellors in a vertical system of control. This seems endemic across the world system, and it would appear exceptions are now possible only where such

systems are absent. The result is the loss of innovative capacity across the world capitalist system. I believe the only solution is to build alternative universities elsewhere.

This requires, although there may be alternatives, distributed and democratic control. In order to maintain resilience under coordinated attack from authoritarian structures, my own

conception considers lowest level authority under structures of coherence. This coherence enables collective investigation of the truth in systems based on authority of reason rather than

direct physical force of authoritarian systems. It implies this coherence is collectively acknowledged under a rolling system of planning under the boundaries of its social group. A wider

system under negotiation of such collectivities, and a wider coherent plan under a general boundary enable the resulting system to innovate without collapsing into authoritarian control.

**Terry Tao** is a very interesting mathematician. I am trying to do the same thing. Although it may seem we come from completely different directions, on reflection this is not the case.

My understanding, which is probably terribly wrong, is that Chinese culture approves of finding explicit results. Tao has good memory, so he can compute. We must assume he does

it accurately. This established his reputation at an early age. I would say he is a problem solver. He does this extremely well. I have very poor working memory. It almost forces me,

despite extensive calculations which at one stage filled 38 refuse bags of calculations, almost all wrong, which I disposed of in a house move, to be a conceptual mathematician. I am not

quite a standard conceptual mathematician. I use intuition and vast analogies. I don't think Tao is unable to do conceptual mathematics, rather the reverse. Mathematics progresses not

only through global conceptual coherence, but also by detailed developments attached to these concepts. It is dependent on its diversity. I think if ever a conceptual mathematician and

a detailed problem solver were to meet in deep thought, the result would push forward mathematics more than the independent research they could develop on their own. This extends not

just to a pair of mathematicians, but the social system in which they are embedded. If the social system were to promote the breaking down of boundaries so that this type of collaboration

happens more often, I think this would be a very substantial contribution to the further development of mathematics. This potential is another reason why I think Terry Tao is interesting.

Return to Table of Contents.

** Number, space and logic** is one half complete, containing fifteen months work. It extends

There will be an exposition and extension of homolgy, cohomology and homotopy theory in it.

John Baez has made accessible a wide area of technical mathematics to a huge audience. It is deservedly popular. He is deeply aware of the contours of the subject, and what is significant.

I do not think he has been the critic of some result in the way I have. This has resulted from different aims and personalities. I have time available by being outside the system. If you have

not already done so, please look at his site. Here is a comment on **John Baez's blog** dated 28-29 June 2017, also the prevention of a discussion of it from a Sussex University administrator.

The curent contents may contain conceptual and other errors.

**VOLUME I: Insight on Computable Superstructures**

1A. ** Insight on Computable Superstructures Foreword and table of contents.**

1B. ** The Meaning of the Finite and the Infinite Chapter 1. **We introduce a modified set theory. We extend the rules for the positive whole numbers to transfinite natural numbers, called

transnatural numbers. Some basic properties of the transnatural numbers are developed. We define a model for the real numbers using 'capital Zeta functions', deconstruct the uncountability

assumption for the set of subsets of the natural numbers and introduce an algebra of transinfinite ordinals, called ladder algebra.

1C. ** The Meaning of Branched Spaces Chapter 2. ** We discuss lattices. Removing a point from a real line divides it into two pieces. So we say removing a point from an n-branched space

divides it into n pieces. We introduce a model for a number descibing branched space topology, the Euler characteristic, as a general polynomial with integer coefficients.

1D. ** The Meaning of Suoperators Chapter 3. ** Just as addition by repetition generates multiplication and multiplication by repetition generates exponentiation, so an nth superexponential

operation, called a suoperator, generates an (n + 1)th suoperator. We introduce a general algebra for suoperators in an extended form from that given in the eBook *Superexponential algebra*.

1E. **Zargonions Chapter 4.** A sequence of algebraic structures begins with real numbers, complex numbers, the quaternions and then octonions. **Adonions** are a generalisation of

octonions, generating algebras with division of dimension 6k + 2 for whole numbers k. This is in violation of the result of J.F. Adams *On the Nonexistence of Elements of Hopf Invariant One*, which uses Steenrod squares, equivalent to using K theory. But K theory does not consider sporadic groups, and this is a gap in the proof. This survey collects together results on

division algebras, the novanions of Superexponential Algebra and their combinations, called zargonions. Additional to the chapter in Superexponential algebra, we show that the 10- and

26-novanions have both fermionic (odd number of twists) and bosonic (even number of twists) representations, and there is an enveloping novanion.

1F. **The Meaning of Categories and Superstructures Chapter 5.** We give a survey of category theory, including a discussion of functors, universals and adjoints. We generalise

to the nonassociative case called superstructure theory, considering limits, colimits, choice, comma categories, Kan extensions, toposes and sutoposes.

1G. **Groups Chapter 6.** We describe basic group theory and prove the three isomorphism theorems. We introduce the Schur multiplier and look at the standard classification of simple

groups. We also prove the orbit-stabiliser theorem, Sylow's theorems and the Jordan-Hoelder theorem. We give a counter-example to the statement that normal groups form a modular

lattice. We describe the classification of Lie algebras and vertex operator algebras.

1H. **The Meaning of Superstructure Games Chapter 7.** We introduce a discussion of games in the context of superstructure theory.

1I. ** The Discovery of the Polynomial Wheel Chapter 8.** Polynomial wheels are general techniques to solve polynomial equations of any degree by radicals, previously thought

impossible. We introduce polynomial wheel theory and comparison methods and survey similar results to Superexponential algebra volume II, demonstrating that the Galois solvability

model is incorrect for comparison methods which avoid killing central terms, show by general methods that solutions of polynomial equations of arbitrary degree can always be found in

principle (this we cal Birkby's theorem. All general arguments of this type can be re-expressed in category theory), provide a geometric realisation of the cubic, and look in detail at the

solvability of quintic polynomials. We review the solvability of the quintic as embedded in a quartic variety in squared variables, showing a link with elliptic curves. The general degree

descent techniques are called polynomial wheel methods and give for example solutions of the quintic by radicals.

1J. ** Consistent Problems are Solvable Chapter 9.** The quintic gives a solution of the sextic. This is part of a general theorem that a solvable polynomial of odd degree n gives rise to

a solvable polynomial of degree n + 1. This process can be iterated indefinitely and is part of a general theorem that all consistent problems are solvable - unsolvable problems are

inconsistent. Solutions by comparison methods are further investigated. We discuss P/NP problems in the context of Birkby's theorem, and generalise the discussion on elliptic curves

of chapter 7.

**VOLUME II: Finite and Infinite Number Theory**

2A. ** Finite and Infinite Number Theory Foreword and table of contents.**

2B. ** Number Theory Chapter 1 is half constructed. ** We look at ladder algebra and its transcendental extensions. The chapter has a section on Gauss, Ramanujan and Kloosterman sums.

2C. ** Class Field Theory Chapter 2 is not ready. **We look at class field theory.

2D. ** Theta Functions Chapter 3 is not ready. **We look at the work of Jacobi and Cayley on theta functions.

2E. ** The Consistency of Analysis Chapter 4 is not ready. **We develop a Gentzen-type proof of the consistency of analysis from our theory of trees and our transcendental results. The

theory is extended to cover colour logics, also those of intuitionalistic type.

2F. ** Fermat's Last Theorem Chapter 5 is half complete. **We prove Fermat's Last Theorem using elementary methods and give a more sophisticated treatment using the modularity theorem.

2G. ** Zeta Functions Chapter 6. **The chapter proves the classical theory of the Riemann hypothesis using invariance under imaginary Dw exponential algebras and will prove the general

Riemann hypothesis by extending w.

2H. ** Eisenstein Series Chapter 7 is not ready.**

2I. ** The Goldbach Conjecture Chapter 8 is not ready. **The 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.

**VOLUME III: Algebra and Logic Combining Trees**

4A. ** Algebra and Logic Combining Trees Foreword and table of contents.**

4B. ** Trees and Amalgams Chapter 1 is not ready.** We define trees and amalgams.

4C. ** Branched Spaces and Explosions Chapter 2 is not ready.** We describe 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.

4D. ** Surgery Chapter 3 is not ready.** We obtain 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.

4E. ** Optimisation and Complexity Chapter 4 is not ready.**

4F. ** Sequent Calculus and Colour Logic Chapter 5 is not ready.** We look at multivalued logics, called colour logics, develop Gentzen's sequent calculus for logical deduction and extend

the discussion of sequents to modal and colour logic.

4G. ** Probability Logic, Distributions and Zargonion Fractals Chapter 6 is not ready.** The chapter looks at probability logics from the point of view of distributions, and how to weight

polynomial curve fitting so that the resulting curve has maximum significance. It revisits colour logic in the probability context as a multidimensional logic, extends this to zargonions, and

then incorporates dimension in terms of zargonions with real coefficients as fractal structures.

4H. ** Glyphs Chapter 7 is not ready.**

**VOLUME IV: Varieties and Topology**

3A. ** Varieties and Topology Foreword and table of contents.**

3B. ** Zargon Varieties Chapter 1 is not ready. **We describe a generalisation of the theory of elliptic curves.

3C. ** Modular Forms - Representation Theory Chapter 2 is not ready. **We include work on modular forms, used in the modularity theorem.

3D. ** Conformal and Nonconformal Analysis Chapter 3 is half ready. **We include work on the hyperintricate Cauchy-Riemann equations and the Cauchy integral formula.

3E. ** Conformal and Nonconformal Suanalysis Chapter 4 is not ready. **This develops suoperator analogues of the previous chapter.

3F. **Riemann Roch Chapter 5 is not complete. **This chapter includes work on the Riemann-Roch theorem, the Gauss-Bonnet theorem and stereographic projection.

3G. ** Homology Superstructures Chapter 6 is not ready. **We describe our replacement of homology in the superexponential context.

3H. ** Cohomology Superstructures Chapter 7 is not ready. **The chapter describes etale and motivic cohomology in the superexponential context.

3I. ** Homotopy Superstructures Chapter 8 is not ready. **We describe homotopy, a theory of paths, in the superexponential context. Epicycle knots are discussed.

3J. **Local Zeta Functions Chapter 9 is not complete.** We look at the Weil conjectures. Implications for the function field case are given, where transl-adic and other results are applied.

**VOLUME V: Sphere Configurations and Simple Groups**

4A. ** Sphere Configurations and Simple Groups Foreword and table of contents.**

5B. **Zargon Lattices Chapter 1.** We discuss lattices from the zargonion point of view. We study the classification of groups, called zargon groups, including simple groups, arising

from the algebra of the zargonions, and investigate the relation of these new results with those obtained by using Dynkin diagrams. We obtain Lie algebras beyond those obtained from

the octonions, with more elements than E8. We obtain some simple groups with similar but different size to the monster, and an infinite sequence of such simple groups. We discuss

implications for quantum mechanics.

5C. **Lattice Theta Functionss Chapter 2 is not complete.**

5D. **Kissing Numbers Chapter 3 is not complete.**

5E. **Hamming and Golay codes Chapter 4 is not complete.**

5F. **The Novanion 9-1 Lorentzian Lattice Chapter 5 is not complete.**

5G. **The Adonion 13-1 Lorentzian Lattice Chapter 6 is not complete.**

5H. **The Novanion and Adonion 25-1 Lorentzian Lattices Chapter 7 is not complete.**

5I. **Moonshine Chapter 8 is not complete.**

5J. **The Adonion 31-1 Lorentzian Lattice Chapter 9 is not complete.**

5K. **Enveloping Zargon Moonshine Chapter 10 is not complete.**

5L. ** 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.

You can **comment** on this eBook, ** Superexponential Algebra**.

**VOLUME I**

0A. ** Remarks**

1B. ** Foreword The Foreword, Table of Contents and Mathematical terms **of Superexponential Algebra Volume I.

1C. ** Prologue Why is mathematics there? and What is mathematics? **of Superexponential Algebra.

1D. ** Intricate Numbers Chapter I** introduces the intricate matrix representation.

1E. ** Hyperintricate Numbers Chapter II** introduces the hyperintricate matrix representation.

1F. ** Associative Division Algebras Chapter III. **We prove 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 give a proof of Wedderburn's little theorem.

1G. ** Nonassociative Algebras Chapter IV. **We introduce nonassociative operations derived from associative matrix multiplication, extend the hyperintricate methodology and discuss Lie and Kac-Moody algebras.

1H. ** Novanions Chapter V. **We prove 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 and Number, Space and Logic.

1I. ** 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.

**VOLUME II**

2A. ** Foreword The Foreword, Table of Contents and Mathematical terms **of Superexponential Algebra Volume II.

2B. ** Ladder and Complex Algebra Chapter VII. **We discuss the incompatibility between the continuum hypothesis and the countability of the rational numbers and introduce 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. ** The 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. ** We show that ring automorphisms for polynomials are commutative, unlike inner group automorphisms, so Galois theory fails.

2F. ** Solvability of Complex Varieties Chapter XI.** The chapter shows polynomial equations are equivalent to varieties in two variables. An 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.

**VOLUME III**

3A. ** Foreword The Foreword, Table of Contents and Mathematical terms **of Superexponential Algebra Volume III.

3B. ** Algorithms and Consistency Chapter XIV.** We look at nonstandard interpretations of the continuum hypothesis and discuss 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.** The superexponentiation operations generalise e^{a}e^{b} = e^{a + b}. Mappings discussed here are not associative.

3F. ** Appendices, Answers to Exercises, References and Index Chapter XVIII.** The concluding part of the eBook.

__Historical background__

The * Mathematical archive* accesses historical material that went into the creation of Superexponential algebra.

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

S_{5} --> S_{4} --> S_{3} --> S_{2} --> 1.

It has been obtained by a formal 'non inertial' differential condition. We describe an assumed method of Grothendieck, which uses multifunctions and in the case of the sextic corresponds to the series

S_{6} --> S_{3} --> S_{2} --> 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 2^{n} X 2^{n} 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 201^{st}! - 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, II and III of the Quadratic Residues papers, investigating and proving 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 solve the problem of the previous two papers, using the average column of the average quadratic residue. We also 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.

__Historical background__

The * Mathematical archive* accesses historical material that went into the creation of Innovation in mathematics.

Return to Table of Contents.

__Other early papers__

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.

__Historical background__

The * Mathematical archive* accesses historical material that went into the creation of Elementary methods in number theory.

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