PROOF OF THE EXISTENCE OF GOD

SCIENTIFIC AND DIRECT PROOF OF the Existence of God

The human predicament is a Self-imposed enigma, born of circular reasoning: It is literally only a self-indulgent quirk of ego that leads serious minded people to the erroneous belief that the only thing that can actually be proved is unprovable, and that the illusion of that belief is provable!

The only thing that you can actually prove is the existence of consciousness. This is because consciousness is the only thing that you experience directly. All other knowledge is obtained very indirectly through the severely limited filter of the physical senses. The illusion of separate existence, on the other hand, is totally based on the fallacious belief that reality can exist without consciousness. This is the very basis of the current scientific paradigm, and yet it is totally un-scientific: A scientific hypothesis, by definition, must be open to proof or disproof. Thus the hypothesis that reality exists, or has ever existed, apart from consciousness, is literally un-provable, because no reality is observable without the existence of a conscious observer.

Once you allow yourself to think outside the self-imposed box of the ego-based illusion of separate existence, the clues of quantum physics and relativity become quite obvious: time and space, mass and energy, mind and matter, are not what you think they are! Their apparent existence as elements of a reality independent of consciousness depends upon the actual existence of your consciousness, and its willingness to indulge in the illusion of confinement to a separate existence.

It is, however, within your power, as a spark of the Reality of Primary Consciousness, to free yourself from identification with the illusion of a separate physical existence. It is time to grasp the key to the human dilemma; find the door in your cage, open it and allow the Eagle of your Spirit to escape the self-imposed confinement of material imprisonment and soar into the Infinite Reality beyond. The new sensation of impending freedom may be frightening, but there is nothing to fear but fear itself. The Way-Shower is always there, waiting for you in whatever form your heart requires, just outside the door, extending a helping hand.

The existence of a reality separate from consciousness is forever unprovable. On the other hand, the existence of Primary Consciousness, sometimes called God, is scientifically provable, once science is freed of the self-imposed constraints of materialism, and directly provable for each and every one of us through the direct experience of Infinity, once your consciousness is freed from the self-imposed illusion of identification with an imaginary ego. The place and time of your Enlightenment and Realization of Who you really are is very real and always available in the Here and Now. It only requires your willingness to drop the seemingly safe, comfortable and intoxicating illusion of ego confinement.

There is no available, or even possible, proof that life and consciousness have emerged from random arrangements of debris flying away from a great explosion. There are, however, many ways to prove the opposite, i.e. that the physical universe is a finite manifestation of Primary Consciousness. We can now prove scientifically that finite reality is an ever-expanding small part of an amazingly complex and beautiful Self-referential Reality, a conscious expression of the awe-inspiring grandeur of a glorious infinite existence, a blissful Reality without beginning or end. 

The idea that the reality we experience is part of an infinite conscious Reality is, of course, not new. What is new, is that we have proved it by putting consciousness into the equations of science for the very first time. See articles on the TRUE units of mass, energy and consciousness, and the e-book  "Reality Begins with Consciousness" on wwwBrainVoyage.com.

A Leading Scientist Speaks Out on Transcendental Physics

Read what one of the most advanced scientific thinkers today, has to say about Transcendental Physics:

“This book could be the most important book of the twentieth century, having more long-term impact on the advancement of the human race than any other. It is the logical integration of every valid scientific principle to date and proves that our universe is both physical and non-physical …

“Dr. Close has accomplished what no one has ever done before. He has taken all the fundamental discoveries and facts of modern science and has proven, conclusively, that none of these laws of science could be true without the all-pervading consciousness of God –be it called Allah, Brahma, Jehova, Great Spirit, Yahweh, Heavenly Father, or Divine Mother… Thanks to this life-long labor by Dr. Close, we can now whole-heartedly practice our faith in God and pursue our science with equal fervor… He has shown us by exquisite and flawless logic, without passing over  or ignoring any aspect of science, that to be a scientist without a belief in God is a contradiction and that it is equally a contradiction to be a spiritual devotee without accepting the discoveries of science.

“One cannot place too much importance on the genius of this work. I, for one, am thankful for Ed Close’s devotion and commitment to this project that has never flagged over the years. Because of his perseverance, a door has been opened for us all. He has broken forever, the barrier between science and religion …

“… Science was confined in a self-imposed cage with the majority of the universe on the outside. Ed Close has broken the bars and forged a key link in the chain of advancement of mankind for the ages. This book should be read by everyone who seeks to know the truth. It is worth every effort to study and comprehend –even if you have to read it several times. It could change your entire outlook on life forever, and change it for the better.

"...take this work to heart, for through it you 'shall know the truth, and the truth shall make you free.' (John 8 : 32)."

--Dr. David Stewart, PhD, Author, Educator, Internationally known Speaker, and Consultant

Fermat's Last Theorem 1965 proof and Update

CLARIFICATION AND EXPLANATION OF CLOSE’S 1965 PROOF OF

FERMAT’S LAST THEOREM

By Edward R. Close PhD

“Concerning whole numbers, while certain squares can be separated into two squares, it is impossible to separate a cube into two cubes or a fourth power into two fourth powers or, in general any power greater than the second into two powers of like degree. I have discovered a truly marvelous demonstration, which this margin is too narrow to contain.” - Pierre de Fermat, circa 1637 [1]

ABSTRACT:

Fermat’s last theorem (FLT) states that no three positive integers X₁, Y₁, and Z₁ can satisfy the equation xⁿ +yⁿ = zⁿ for any integer value of n greater than two. Though a proof was claimed by Pierre de Fermat in 1637, it was never found and Fermat’s Last Theorem remained officially without proof until Andrew Wiles published an extensive solution in 1995. Prior to that, however, Edward Close in 1965 provided a solution and submitted a proof, which he then published in 1977 (FLT65). The Close proof though never refuted presented difficulties for reviewers because of the unconventional notation used and three reviewers have suggested that the difference between algebraic polynomial factors and integer factors as used made the proof unclear and possibly incorrect. In this paper, Close provides FLT65C as a clearer rewrite of FLT65. He also demonstrates that FLT65 provided a true and adequate proof of FLT. The FLT65 proof remains unchanged in this presentation but adherence to conventional notation makes it clearer, and the polynomial critique is shown to be irrelevant in this rewrite (FLT65C). Additionally, the fact that the case of the exponent N=2 is excluded from the proof is shown more clearly. This paper provides a 2-page FLT65C solution, and then in the discussion amplifies and clarifies areas that may help reviewers understand the proof. Several pertinent appendices are attached. Appendix A demonstrates the proofs of the division algorithm and corollaries because they are central to the proof. Additionally, the application of primes and relative primes is an important part of the proof of FLT65: Appendix B shows the proof of sufficiency for X, Y and Z relatively prime and N = primes > 2, in any proof of FLT. And importantly, Appendix C provides the original FLT65 published proof.

INTRODUCTION

Sometime around 1637, Pierre de Fermat scribbled a brief statement in Latin in the margin of a book on Diophantine equations. This theorem came to be known as “Fermat’s Last Theorem” (FLT), because it remained without a known proof for more than three centuries. In modern representation, FLT is stated as follows: No three positive integers X₁, Y₁, and Z₁ can satisfy the equation xⁿ +yⁿ = zⁿ for any integer value of n greater than two. The first generally accepted proof has been attributed to Sir Andrew Wiles, who is said to have spent seven years completing it. Dr. Wiles’ proof was published in 1995.

In 1965, three decades before Wiles’ proof was announced, I (Edward R. Close) had produced a proof of FLT. At the time, I had completed my BA degree in mathematics with a minor in physics, one semester of a Master’s program in theoretical physics, and had been teaching mathematics for three years. The 1965 proof was submitted to a mathematician at Iowa State University. He returned it with a note saying that “It could not be a valid proof because it would apply to the case n = 2”. From his response, I knew he had not read past the first page, because the case n = 2 was eliminated on the second page. I learned that most professional mathematicians consider reviewing a purported proof of FLT a waste of time, and look for anything that appears to be an error and reject the “proof” as quickly as possible. I was very discouraged and went on with my life, but later published my proof anyway in an appendix of “The Book of Atma”. ^[2] Over the years, my 1965 proof of Fermat’s Last Theorem (FLT65) has been submitted to a number of professional and amateur mathematicians for review and comment. The responses from those to whom the proof was submitted fall into one of the following categories:

1.      Some declined to look at it either because they didn’t have the time or they didn’t want to take the time to review it.

2.      Some refused to look at it because number theory was not their field of expertise and/or interest.

3.      Some refused to look at it because they did not believe a simple proof of FLT possible.

4.      Several of the reviewers found nothing wrong with the proof.

5.      Some of the reviewers found the argument inconclusive.

Those who found the argument inconclusive cited one or both of the following reasons:

i.        They found the notation unclear or confusing.

ii.      They were concerned that conclusions drawn from the factorization of the FLT equation, xⁿ + yⁿ = zⁿ, as a polynomial in n, might not apply to the numerical factorization of the polynomial for some specific integer solution (X₁, Y₁, Z₁), providing a potential loop-hole in the proof. [1]

The notation in the 1965 proof may have confused readers because it is non-standard and not as well defined as it should have been. At the time, I used unconventional notation deliberately because I thought it made it easier to distinguish when the arguments were being considered as variables from when they were being considered as specific integers. The subscript identified an algebraic symbol as an integer, as opposed to a variable. I failed, however, to provide a notation that distinguished between integer variables and specific integer constants, and this was problematic because confusion may arise if a specific notation is not clearly defined and its use properly justified. Therefore, in this discussion, I will use three types of notation precisely defined for greater clarity:

·         Lower-case letters, like x, y and z represent variables with no numerical restrictions.

·         Upper-case letters like X represent variables restricted to integers.

·         Upper-case letters with subscripts like X₁ represent specific integer values of the variables.

In addition, we will distinguish between integer factors and algebraic factors as follows:

·         g(x) e f(x) means the polynomial g(x) is an algebraic factor of the polynomial f(x), or stated another way, g(x) is contained in f(x) as an algebraic factor; and

·         A ∈ B means A is an integer factor of the integer B, or A is contained in B as an integer factor.

·          Consistent with ≠, meaning “is not equal to”, the oblique strike through a symbol will indicate the negation of the symbol; e.g.: g(x) ɇ f(x) means g(x) is not an algebraic factor of the polynomial f(x) and A ∉ B means A is not a factor of B.

My position has been, and remains, that FLT65, my 1965 proof, is conclusive and complete, even though the notation may be confusing. I maintain this position for two simple reasons: (A) The division algorithm and corollaries applied in the proof are valid over the entire field [2] of real numbers, which includes the integers, and (B) Application of the division algorithm to the FLT equation produces a unique remainder that allows us to determine the numerical type of the third variable when two of them are assumed to be integers.

Reviewers in category 5 above, with concern number ii, thought that conclusions drawn from the factorization of the FLT equation, xⁿ + yⁿ = zⁿ, as a polynomial in z, might not apply to the numerical factorization of the polynomial for some specific integer solution (X₁, Y₁, Z₁). While this is a legitimate concern, it is dispelled by detailed clarification of notation and step-by-step explanation of the reasoning.

After this introduction, I will present the mathematics of FLT65 with no more discussion than is necessary to be complete. Then I will present a more detailed discussion and explanation of the proof, showing how unique features of the FLT equation provide proof that X, Y and Z cannot all three be integers, and how this eliminates concern number ii above.

The Division Algorithm and three Corollaries, presented briefly below, are central to this proof, and their proofs were included in FLT65. In order to make this presentation as complete as possible, these supplementary proofs are included in Appendix A.

The division algorithm states that if g(x) ≠ 0 and f(x) are any two polynomials over a field, of degree m and n, respectively, and n > m, then there exist unique polynomials q(x) and r(x) such that f(x) = q(x)g(x) + r(x), where r(x) is either zero or of degree smaller than m.

Corollary I states that if f(x) and g(x) contain a common factor, r(x) contains it also.

Corollary II states that the remainder, r(x), when f(x) is divided by (x-a) is f(a).

Corollary III states that a polynomial, f(x), of degree greater than one x-a, IF AND ONLY IF, f(a) = 0.

To prove FLT, it is necessary to show that for n ≥ 3, there is no solution (X₁,Y₁, Z₁) such that X₁, Y₁ , and Z₁ are integers. It is necessary and sufficient to show this for n = p, where p is a prime number and X₁, Y₁ and Z₁ are relatively prime. Proofs of these two statements are given in the 1965 proof. They are relatively simple and well-known. For efficiency they are not included in the proof below, but for completeness and clarity, they are included in Appendix B. The method employed in the 1965 proof of FLT, is to show that if we restrict x and y to the field of integers, z cannot be an integer. Clearly, when this is shown, FLT is proved.

FERMAT’S LAST THEOREM (FLT)

(A TWO PAGE PRESENTATION OF THE CLOSE 1965 PROOF [FLT65C])

Consider the equation z^p – x^p = y^p. …… equivalent to Equation (1) in the 1965 proof. Since p is a prime number > 2, and thus an odd prime, we can factor the left side of the equation to obtain:

(z-x)( z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = y^p…… equivalent to Equation (2) of FLT65C.[3]

For variable integer values of x, represented by X, let z^p-1 + z^p-2X + z^p-3X² +•••+ zX^p-2 + X^p-1 = f(z),

and z-X = g(z). Then g(z)f(z) = Y^p, for all integer values X and Y. …… Equation (3) of FLT65C

For Fermat’s last theorem to be falsified, X, Y and z must be integers, so we will replace X and Y with X₁ and Y₁, representing specific integers. But, since we do yet not know whether z can actually be an integer if x and y are integers in an FLT solution, we must continue to represent it by z, a variable over the field of real numbers.

By Corollary I of the DIVISION ALGORITHM, since f(z) and g(z) are polynomials in z of degree n = p -1 and m = 1, respectively, when f(z) is divided by g(z), the remainder, r(z), will contain any and all algebraic factors common to both.

And by COROLLARY II, the remainder when f(z) is divided by g(z) will be f(X₁). And so:

r(z) = f(X₁).= X₁^p-1 + X₁^p-2X₁ + X₁^p-3X₁² +•••+ X₁^p-1 = pX₁^p-1, a unique constant made up of integer factors for any p > 2. … Equation (4.).

Since for any solution of Equation (1), X₁, Y₁ and z, if an integer, may be considered to be relatively prime, no factor of X₁, and, therefore, of X₁^p-1, may be contained in g(z) = z –X₁. Therefore, if f(z) and g(z) have a common factor, it must be p. It also follows that for their product to be equal to the perfect p-power, (Y₁)^p, either p ∈ f(z), p ∈ g(z), or they must both be perfect p-powers of integers.

By exactly the same reasoning as above, we can write the FLT equation as z^p – y^p = x^p and factor as:

(z-Y)( z^p-1 + z^p-2Y + z^p-3Y² +•••+ Y^p-1) = X^p. … Equation (5)

Let z^p-1 – z^p-2Y₁ + z^p-3Y₁ ² -•••+ Y₁ ^p-1 = f₁(z) and z - Y₁ = g₁(z).

Then, dividing f₁(z) by g₁(z), analogous to Equation (4), we obtain r₁(z) = pX₁^p-1… Equation (6)

Then, either p ∈ f₁(z) and p ∈ g₁(z), or they are perfect p-powers of integers.

For a given case of z^p = X₁^p + Y₁^p, it is possible that either p ∈ f(z) or p ∈ f₁(z). But, if p is a factor of one, the other has to be a perfect p-power, since X₁ and Y₁, and therefore, f(z) and f₁(z) are relatively prime. For a given z = Z₁, either p ∈ f(Z₁) → p ∈ Y₁, or p ∈ f₁(Z₁) → p ∈ X₁. If neither X₁.nor Y₁ contains p, because they are relatively prime, both must be perfect p-powers. Therefore, in any event, one of them at least, must be a perfect p-power, not containing p as a factor.

Since f(z) and f₁(z) are both of the same form, we may choose either one or the other as not containing p. So we may choose f(z) = z^p-1 + z^p-2X₁ + z^p-3X₁² +•••+ X₁^p-1 = A^p, with A ∉ p.

By the Division Algorithm, for the two polynomials, g(z) ≠ 0 and f(z), over the field of real numbers, with degrees m and n respectively, and n > m > 1, there exist unique polynomials q(z) and r(z) such that f(z) = q(z)g(z) + r(z), where r(z) is either zero or of degree smaller than m.

If n = m = 1, as in the case when p = 2, or if f(z) is not equal to an integer raised to the p^th power, as in that case when A is not an integer, but is the p^th root of a prime number, q(z) and r(z) are not unique and COROLLARY II does not hold. Thus, this proof does not apply to the case n = 2, or to non-integer solutions of the FLT equation. But when p > 2, and we assume there is an integer solution of equation (1), because the set of real numbers is closed with respect to addition, for any value of z, there is some real number s, such that z – s = A, and COROLLARY II tells us that if g(z) = z – s, a polynomial of degree 1 in z, q(z) and r(z) are unique and the remainder, r(z) will be of degree m < 1 = zero degree, and of the form f(s). Therefore: f(z) = (z-s)q(z) + f(s) over the field of real numbers, and by COROLLARY III, f(z) e (z – s), IF AND ONLY IF, f(s) = 0. Thus when p >2, we have:

(z – s) e f(z) → f(s) = s^p-1 + s^p-2X₁ + s^p-3X₁² +•••+ X₁^p-1 = 0….Equation (7).

Since both s and z can take on any real number value, if they are not integers, f(s) = 0 is not a contradiction because there are an infinite number of real number solutions for the FLT equation. But the Division Algorithm and Corollaries hold over the field of real numbers, including integers, so for specific integer values z = Z₁, s = S₁ and A = A₁, (z – s) e f(z) →(Z₁ – S₁) ∈ f(Z₁) → f(S₁) = 0.

But f(S₁) = S₁^p-1 + S₁^p-2X₁ + S₁^p-3X₁² +•••+ X₁^p-1 = 0 is an impossibility because X₁ and S₁ are positive integers, and the sum of positive integers cannot equal zero.

Therefore, for specific integer values z = Z₁, s = S₁ and A = A₁, (Z₁ – S₁) ∈ f(Z₁) → f(S₁) ≠ 0, and thus

g(z) = (Z – S) ∉ f(z) = z^p-1 + z^p-2X₁ + z^p-3X₁² +•••+ X₁^p-1.

But, for there to be a triple integer solution for the FLT equation, falsifying FLT, there must be specific integers X₁, Y₁, Z₁, A₁, and S₁, such that (Z₁ – S₁) = A₁, and f(Z₁) = A₁^p, therefore:

(Z₁ – S₁) ∉ f(Z₁) → A₁ ∉ A₁^p and we have a clear contradiction, proving Fermat’s Last Theorem (FLT65).

DISCUSSION AND EXPLANATIONS

OF THIS PROOF OF FERMAT’S LAST THEOREM (CLOSE FLT65C)

This is the same proof FLT65C but with detailed discussions and explanations:

Using the notation outlined in the introduction, we can write: z^p – x^p = x^p. This is equation (1) in the 1965 proof, without the restriction of x, y and z to integer values.

Since p is a prime number, we can factor the left side of the equation to obtain

(z-x)( z^p-1 + z^p-2x + z^p-3x² +•••+ x^p-1) = y^p, equation (2) of the up-dated 1965 proof

In the 1965 proof (FLT65), the FLT equation is expressed as an N-degree polynomial in the variable Z. This translates to a p-degree polynomial in the variable z in the current standardized notation. The rationale for this approach to proving FLT by focusing on one of the variables as an independent unknown is based on the fact that any equation in three unknowns, including the FLT equation, has an infinite number of solutions, but, if we arbitrarily choose values for two of the variables, we can solve for the third. In a similar manner, if two of the variables of the FLT equation are restricted to the field of integers, we will be able to determine whether any values of the third variable can be integers.

In FLT65, Close focused on z as the independent unknown by setting X = X₁ and Y = Y₁, intending only to imply that they were integer variables. Later on, I indicated that they were specific integer values, without changing the way they were represented. This lack of clear definition of notation is at least part of the confusion that gave rise to the concern #ii.

My thinking was that taking the trouble to distinguish between variables and specific values of the variables was unnecessary because the division algorithm and corollaries apply to both, as ultimately, whatever their values, they are elements of the field of real numbers. Reviewers, however, have pointed out that this is not necessarily true for polynomials in general. It turns out to be the case in this instance, however, because of the unique form of the FLT equation and the requirement that the only solutions being considered are those for which all three variables have integer values.

Because at least three of the most qualified reviewers of FLT65 raised the concern about the applicability of the algorithm and corollaries to factors of integers, it is clear that the proof is difficult to follow unless one proceeds through the whole process with clearly defined and justified notation. When this is done below, we see that the legitimate application of the algorithm and corollaries to variable polynomials derived from the FLT equation leads to an unavoidable contradiction that proves FLT.

In this more detailed explanation, we will adhere to the notation defined above, viz. we will use x, y and z for unrestricted variables over the field of real numbers, X, Y and Z for variables restricted to the sub-set of real numbers that make up the ring[4] of integers, and X₁, Y₁, and Z₁ for specific integers; and for clarity, we will distinguish between integer factors and algebraic factors as follows:

g(x) e f(x) means the polynomial g(x) is an algebraic factor of the polynomial f(x), or stated another way,

g(x) is contained in f(x) as an algebraic factor; and A∈ B means A is an integer factor of the integer B, or A is contained in B as an integer factor. Also, consistent with ≠, meaning “is not equal to”, the oblique strike through will indicate the negation of negation of the symbol; e.g.: g(x) ɇ f(x) means g(x) is not an algebraic factor of the polynomial f(x) and A ∉ B means A is not a factor of B.

For variable integer values of x, represented by X, let z^p-1 + z^p-2X + z^p-3X² +•••+ zX^p-2 + X^p-1 = f(z)

And z-X = g(z). Then

(3.)       g(z)f(z) = Y^p, for all integer values X and Y.

For Fermat’s last theorem to be falsified, X and Y must be specific integers, call them X₁ and Y₁, and z must also be an integer. But, since we do not yet know whether z can actually be an integer in the FLT equation, we must continue to represent it by z, a variable over the field of real numbers.

By Corollary I, since f(z) and g(z) are polynomials of degree n = p -1 and m = 1, respectively, over the field of real numbers, when f(z) is divided by g(z), the remainder, r(z), will contain any and all factors common to both. See Appendix A for the proof of this.

And by COROLLARY II, the remainder when f(z) is divided by g(z) = z - X₁, will be f(X₁). And so:

(4.)       r(z) = f(X₁).= X₁^p-1 + X₁^p-2X₁ + X₁^p-3X₁² +•••+ X₁^p-1 = pX₁^p-1, a unique integer for any p > 2. Interestingly, it is this unique integer remainder in the case of the FLT equation that will allow us, to extend the application of the division algorithm from variable polynomials over the field of real numbers to integers, a subset ring of the field of real numbers.

Since for any integer solution of equation (1), X₁ and Y₁ may be considered to be relatively prime, no factor of X₁, or, therefore, of X₁^p-1, may be contained in g(z) = z –X_1, because, if z is to be an integer, z –X₁ must contain a factor of Y₁. Therefore, if f(z) and g(z) have a common factor, it must be p. It also follows that for their product to be equal to the perfect p power, (Y₁)^p, one of them, i.e. either f(z) or g(z), must contain p or they must be both perfect p-powers of integers.

Similarly, we can factor the FLT equation as

(5.)       (z-Y)( z^p-1 + z^p-2Y + z^p-3Y² +•••+ Y^p-1) = X^p. And by exactly the same reasoning as above, for any particular Y = Y₁, we can have

z^p-1 – z^p-2Y₁ + z^p-3Y₁ ² -•••+ Y₁ ^p-1 = f₁(z) and z - Y₁ = g₁(z).

Then, either f₁(z) and g₁(z) contain p as a single common factor, or they are perfect p-powers of integers. So for a given case of z^p = X₁^p + Y₁^p , if either f(z) or f₁(z) contains p, the other has to be a perfect p-power, since we have concluded that we only have to consider relatively prime X, Y and Z, implying both cannot contain p. Now, p e f(z) → p ∈ Y₁ and p e f₁(z) → p ∈ X₁. But X₁.and Y₁ are relatively prime. If neither X₁.nor Y₁ contains p, both, being relatively prime, must be perfect p-powers. Therefore, in any event, one of them at least, must be a perfect p-power, not containing p as a factor.

Therefore, we may choose f(z) = z^p-1 + z^p-2X₁ + z^p-3X₁² +•••+ X₁^p-1 = A^p, and/or

f₁(z) = z^p-1 – z^p-2Y₁ + z^p-3Y₁ ² -•••+ Y₁^p-1 = B^p, A and B integers, and at least one, A or B, does not contain p. Also note that since g(z)f(z) = Y₁^p, and f(z) = A^p, A ∈ Y₁ and since g₁(z)f₁(z) = X₁^p, f₁(z) = B^p, B ∈ X₁.

Since f(z) and f₁(z) are both of the same form, we may choose either of them as the one not containing p. So we may choose f(z) = z^p-1 + z^p-2X₁ + z^p-3X₁² +•••+ X₁^p-1 = A^p, and A will not contain p.

We cannot, at this point, assume that z = Z₁, a specific integer, for some x = X₁ and y = Y₁, because this cannot be justified unless we can show that the requirement that f(z) = A^p does not lead to a contradiction.

For any value of X₁, the conditions f(z) = A^p, and A ∈ f(z) are necessary conditions for an integer solution of the FLT equation to exist. And since A is a positive integer variable, with any specific X₁ < Y₁ < any z = Z₁ that will satisfy the FLT equation, we can set A = z – S, where z is an unrestricted variable, and S is a positive integer variable. Note that S is a variable over the ring of integers. Because we don’t know whether z can be an integer, the specific value of S, S₁, when X = X₁, is dependent on the specific integer value, A₁, of A.

The division algorithm tells us that for two polynomials, g(z) ≠ 0 and f(z), over the field of real numbers, with degrees m and n respectively, and n > m > 1, there exist unique polynomials q(z) and r(z) such that f(z) = q(z)g(z) + r(z), where r(z) is either zero or of degree smaller than m.

 Notice that if n = m = 1, as in the case when p = 2, or if f(z) is not equal to an integer raised to the p^th power, as in that case when A is not an integer, but is the p^th root of a prime number, q(z) and r(z) are not unique and COROLLARY II does not hold, and this proof does not apply to the case n = 2, or to non-integer solutions of the FLT equation. But when p > 2, and we assume there is at least one solution of equation (1) where x, y and z are equal to integers, COROLLARY II tells us that if g(z) = z – s, which is a polynomial of degree 1 in z, q(z) and r(z) are unique and the remainder, r(z) will be of degree m < 1 = zero degree, i.e., a constant, of the form f(s).

Therefore: f(z) = (z-s)q(z) + f(s) over the field of real numbers, and by COROLLARY III, f(z) is divisible by z – s, IF AND ONLY IF, r(z) = f(s) = 0. Thus when p >2, we have:

(6.)       (z – s) e f(z) → f(s) = s^p-1 + s^p-2X₁ + s^p-3X₁² +•••+ X₁^p-1 = 0.

Since both s and z can take on any real number value, if they are not integers, f(s) = 0 is not a contradiction and there are an infinite number of real number solutions for the FLT equation. But, if s is to be from the ring of integers, f(S) = 0 is a contradiction.

Since the Division Algorithm and Corollaries hold over the field of real numbers, including integers, for specific integer values z = Z₁, s = S₁ and A = A₁, (z – s) e f(z) →(Z₁ – S₁) ∈ f(Z₁) → f(S₁) = 0.

But f(S₁) = S₁^p-1 + S₁^p-2X₁ + S₁^p-3X₁² +•••+ X₁^p-1 = 0 is an impossibility because X₁ and S₁ are positive integers, and the sum of positive integers cannot equal zero. Therefore:

g(z) = (z – S) cannot be a factor of f(z) = z^p-1 + z^p-2X₁ + z^p-3X₁² +•••+ X₁^p-1. Furthermore, because the division algorithm and corollaries apply across the field of real numbers, including the integers, it follows that, for any real values of z, S and A, for there to be a triple integer solution for the FLT equation, falsifying FLT, there must be specific integers X₁, Y₁, Z₁, A₁, and S₁, such that z = Z₁ , S = S₁ and A = A₁, and for the FLT equation, (z – S₁) = A and f(z) = A^p, therefore:

(z – S) ɇ f(z) →(Z₁ – S₁) ∉ f(Z₁) → A₁ ∉ A₁^p and we have a clear contradiction, proving FLT.

Note that f(z) and g(z) are polynomials in the variable z throughout the entire discussion, up to the contradiction S₁^p-1 + S₁ ^p-2X₁ + S₁^p-3X₁² +•••+ X₁^p-1 = 0; and also note that because the division algorithm and corollaries apply across the field of real numbers, including the integers, we justified substituting integers, that have to exist in order for FLT to be falsified, into the algebraic forms to obtain the integer factors for the hypothetical triple integer solution of the FLT equation.

When the author did this, he found that the contradiction obtained by assuming that for given integer values of x and y, an integer z was possible also applied to the integer factors as well as the algebraic factors. Therefore the concern that the factorization of X^N + Y^N = Z^N, as a polynomial in Z, might not provide a contradiction in the numerical factorization of the polynomial for some specific integer solution, i.e., concern #ii is not relevant, and the proof of FLT is complete. However, if the reader is not convinced by this argument, we can address concern #ii in additional detail as follows:

Given f(S) ≠ 0, the fact that the integers form a ring, which is a sub-set of the field of real numbers, but technically not a field because the integers are not closed with respect to division, might lead us to believe that for some Z₁ that might satisfy the FLT equation along with X₁ and Y₁, the remainder r(S₁) = f(S₁) = S₁^p-1 + S₁^p-2X₁ + S₁^p-3X₁² +•••+ X₁^p-1, not being zero, might contain g(z) = (z – S₁) = A₁ as a factor. Assuming that there is such an actual integer triple solution for the FLT equation, the integers X₁,Y₁, Z₁, and p might be so large that it would take a million years for the fastest computer available to search for and find this integer solution. The point is that it is possible that there could be an integer solution that we could never find by trying endless combinations of integers from the infinite ring of relatively prime integers. But we can set up a process of infinite descent⁴ by referring to the simple process of long division.

In the process of long division, first we estimate the quotient. Next we multiply our estimate by the divisor, and then subtract the result from the dividend. If the remainder is greater than the divisor, we increase the quotient estimate and repeat the process until the remainder is either zero or less than the divisor. We can follow this same simple procedure with f(S) as the dividend, g(S) as the divisor and r(S) as the remainder, where S is an integer variable over the ring of integers from which the specific integers of a specific solution of the FLT equation must come if FLT is to be falsified.

Because the ring of integers is closed with respect to addition, for every value of S, there is some integer value of K, such that g(S) = S – K = A. Since the remainder, r(z) = f(S) ≠ 0, let’s assume it contains the divisor, A = S - K, i.e., (S – K) e f(S), as it must be for concern #ii to have any validity. Remembering that A, Z and S must all be integers for FLT to be falsified, and due to the well-ordered nature of the ring of integers, allowing for the basic operations of addition and subtraction, there is some (X,Y,Z) = (X_i,Y_i,Z_i), specific integers, such that A = Z_i – S and there is also some specific integer K, such that A = Zi – S = S – K. Then dividing f(S) by S – K, in accordance with the Division Algorithm, we get a remainder equal to f(K):

(7.)       f(K) = K^p-1 + K^p-2X_i + K^p-3X_i² + ••• + X_i^p-1

Since S – K = A, and A has to be a positive integer in order for FLT to be falsified, K < S, and f(K) < f(S). Just as in simple long division, we can repeat this process with A = K – K₁, K₁ < K, and with smaller and smaller K_i until the remainder, f(K_i) obtained is either zero or smaller than A. No matter how large or small the integers of the FLT falsifying solution are, and how large the remainder f(S) may be, the process of infinite descent [5]obtained by successively dividing by A will eventually reduce f(K_i) for that solution to its smallest possible integer value. Now, in order for FLT to be falsified, f(S) must contain A, and for FLT to be falsified, the smallest integer value of f(K_i) must be equal to zero. In our infinite descent, the smallest possible f(K_i) will occur when K_i =1.

But, the smallest possible f(K_i) = f(1) = 1 + X_i + X_i² + ••• + X_i^{p- 1}, which is still a sum of positive integers, and thus cannot equal zero, or contain A, as it must for FLT to be falsified. This constitutes an infinite descent resulting in a contradiction. Assuming that for x^p + y^p = z^p, x = X₁ and y = Y₁, X₁ and Y₁ specific integers, we reach the contradiction of an infinite descent, proving that z ≠ Z, Z ≠ Z₁, and f(z) ≠ A^p, a perfect p^th power of an integer, and all sufficient and necessary conditions for the definitive proof of FLT have been met.

Therefore, we may state unequivocally that the equation x^p + y^p = z^p has no solutions in positive integers when p is an integer > 2, and the proof of Fermat’s last theorem is complete.[6]
APPENDIX A:

PROOFS OF The Division Algorithm and Corollaries

The Division Algorithm

THEOREM: (The division algorithm) If g(x) ≠ 0 and f(x) are any two polynomials over a field, of degree m and n, respectively, and n>m, then there exist unique polynomials q(x) and r(x) such that

(A)       f(x) = q(x)g(x) + r(x)

Where r(x) is either zero or of degree smaller than m.

Let f(x) = a_nxⁿ + a_n-1x^n-1 +•••+ a₁x + a₀

And g(x) = b_mx^m + b_m-1x^m-1 +•••+ b₁x + b_0, b_m ≠0.

If q(x) is zero or of a degree smaller than m we have

f(x) = 0•g(x) + f(x)

and if n = m, q(x) becomes a constant, Q, and r(x) may be of any degree from zero to n, depending on the value of the constant Q. Thus q(x) and r(x) are not unique in this case.

Now we can form f₁(x), of lower degree than f(x), by writing

(B)       f₁(x) = f(x) - a_n/a_m x^n-m •g(x).

We may now complete the proof of this theorem by induction: Assume the algorithm true for all polynomials over the field. Since f₁(x) is such a polynomial, we may write:

(C)   f(x) = q₁(x)g₁(x) + r₁(x),

where r₁(x) is either zero or of a degree less than m. So from (B) and (C):

f(x) = | a_n/a_m x^n-m + q₁(x) | g₁(x) + r₁(x).

or f(x) = Q(x)g(x) + R(x), the desired form of f(x). All that remains is to prove that Q(x) = q(x) and R(x) = r(x), that is, that q(x) and r(x) are unique when m>n.

Now, f(x) = q(x)g(x) + r(x)

And f(x) = Q(x)g(x) + R(x).

This implies Q(x)g(x) + R(x) = q(x)g(x) + r(x), or [Q(x) – q(x)]g(x) = r(x) – R(x).

If m = n, the right member of this equation is either zero or of degree less than m, the degree of g(x). Hence, unless Q(x)– q(x) = 0 → Q(x) = q(x) and r(x) – R(x) = 0 → r(x) = R(x), the left member of the equation will be of a higher degree than m, and we have a contradiction. Thus Q(x) = q(x) and r(x) = R(x), which means that q(x) and r(x) are unique, and the division algorithm is proved over the field of real numbers.

Corollary I

If f(x) and g(x) contain a common factor, r(x) contains it also.

This follows by inspection of equation (A):

            f(x) = q(x)g(x) + r(x) → r(x) = f(x) - q(x)g(x).

Let the common factor be represented by M. Then f(x) = M∙f₁(x) and g(x) = M∙g₁(x), so that r(x) = M∙f₁(x) - q(x)∙M∙g₁(x) = M∙[f₁(x) - q(x)g₁(x)] → r(x) = M∙(a function of x), QED.

Note that this proof holds for M constant or variable, integer or polynomial factor.

COROLLORY II:

The remainder when a polynomial is divided by (x-a) is f(a):.

This follows at once from the division algorithm:

Let g(x) = x-a. Then (A) becomes

            f(x) = (x-a)q(x) + r(x).

By substitution,

f(a) = (a-a)q(a) + r, where r is an element of the field

or, f(a) = r,

so that f(x) = (x-a)q(x) + f(a), QED.

Notice that if q(x) and r(x) are unique, f(a) cannot contain x-a, and so it follows that

COROLLORY III: A polynomial, f(x), of degree greater than one is divisible by x-a IF AND ONLY IF, f(a) = 0.

Note that the division algorithm and all three corollaries hold when f(x) = f(X)and g(x) = g(X), expressions involving only integers, since it is valid over the entire field of real numbers, and integers are elements in the field of real numbers.

APPENDIX B:

PROOF OF SUFFICIENCY FOR X, Y AND Z RELATIVELY PRIME

AND N = PRIMES > 2, IN ANY PROOF OF FLT

X, Y AND Z RELATIVELY PRIME:

When developing a proof of FLT, in the FLT equation X^N + Y^N = Z^N, X, Y and Z may be considered as three relatively prime integers.

X, Y and Z may be considered to be relatively prime because if two of them, say X and Y, contain a common factor or factors, M, then Z contains M also. Proof:

If X = MX₁ and Y = MY₁, then Z^N = (MX₁)^N + (MY₁ )^N = M^N(X₁^N + Y₁^N). And thus Z = MZ₁ , i.e. Z contains M. Also, X^N + Y^N = Z^N → (MX₁)^N + (MY₁)^N = (MZ₁)^N. Factoring M^N out, we have X₁^N + Y₁ ^N = Z₁^N, with X₁, Y₁ and Z₁ relatively prime.

This demonstration proves that any case of the FLT equation with X, Y and Z not relatively prime can be reduced to a case wherein they are relatively prime, and thus if we prove the theorem for X, Y and Z relatively prime, no non-relatively prime case can exist.

N RESTRICTED TO PRIME NUMBERS > 2 IN FLT PROOF:

Definition: a prime number is any integer that is only divisible by itself and 1. The first prime number after unity, and the only even prime number, is 2. But when N = 2, the equation X^N + Y^N = Z^N is known as the Pythagorean Theorem equation, which has an infinite number of integer solutions known as the Pythagorean triples. For example: 3² + 4² = 5². This is why Fermat stated the theorem in the way he did. Translated from Latin, it reads:

“Concerning whole numbers, while certain squares can be separated into two squares, it is impossible to separate a cube into two cubes or a fourth power into two fourth powers or, in general any power greater than the second into two powers of like degree. I have discovered a truly marvelous demonstration, which this margin is too narrow to contain.” Pierre de Fermat, circa 1637 ^[1]

If we can prove FLT for N equal to prime numbers greater than 2, i.e. N = p > 2, we will have proved FLT for all N. Proof:

Let N = ab, with B prime and A any other prime or composite of primes. Then

X^ab + Y^ab = Z^ab → (X^a)^b + (Y^a)^b =(Z^a)^b

This is clearly a case of the FLT equation with N prime. We can assume XYZ ≠ 0, to eliminate trivial solutions that are obtained when one of the triples equals zero, and as shown above, we can assume that x,y,z are relatively prime (sometimes called co-prime).

Since we know that the equation has integer solutions when N = 2, we must consider the cases N = 2a:

X^2a + Y^2a = Z^2a → (X^a)² + (Y^a)² =(Z^a)², which is a Pythagorean equation. Since we know that the Pythagorean equation has integer solutions, the question here becomes: can all three members of a Pythagorean triple be powers of integers. Fortunately, we can eliminate all even powered N from a proof of FLT as follows:

 If a = 2, we have: (X²)² + (Y²)² = (Z²)².

Solution triples for this Pythagorean Theorem equation may be obtained using the well-known formulas for Pythagorean triples. Note:  Close published his derivation of the ratio formulas for the Pythagorean triples in 1977. The formulas were derived from the FLT equation when n = 2 using the properties of real numbers. ^[3]

Using the formulae, we know that there must be two relatively prime integers, P and Q, with PQ > 0, such that:

X² = 2PQ
Y² = P² - Q² → Y² + Q² = P²
Z² = P² + Q²

Now with this application of the formulae, we have obtained another Pythagorean triple:

Y² + Q² = P². By inspection of the equations for X², Y² and Z² above, we see that P < Z, Q < X, and Y < Y². Thus, by assuming that a triple integer non-zero solution exists for the equation (X²)² + (Y²)² = (Z²)² we can produce another integer triple solution with smaller integers. Repeated applications of the formulae will produce smaller and smaller triples, leading to contradiction by infinite descent: Since any integer solution will lead to a smaller integer solution, the smallest integer of the smallest triple must eventually equal the smallest non-zero integer, 1, yielding a triple (1,C,D), where C and D are positive integers, such that (1²)² + (B²)² = (C²)²→ (C²)² - (B²)² = 1. Given that the smallest integer of the successive non-zero triples in the infinite descent will eventually reach unity, and B > A > 1, we can demonstrate the universality of the contradiction as follows: Let C = 3, and D = 2, the smallest positive integers larger than 1. Then (3²)² - (2²)² = 1 → 81 – 16 = 1, which is a clear contradiction, and any pairs of larger integers C₁ > C and D₁ > D, will lead to larger discrepancies. So we have to conclude that there can be no integer solution triples for the equation (X²)² + (Y²)² = (Z²)².

This, of course, is proof of FLT for N = 4, which was first proved by Fermat³. However, this has greater significance than just a proof for N = 4, because any case of X^2a + Y^2a = Z^2a when a is an even number, is a case of N = 4: If a = 2m, m a positive integer, X^2a + Y^2a = Z^2a → X^4m + Y^4m = Z^4m → (X^m)⁴ + (Y^m)⁴ = (Z^m)⁴. Thus the proof of FLT for N = 4 is proof of FLT for all even N, and since all non-prime odd integers are factorable into prime numbers, it is sufficient to consider N = p, a prime in any proof of FLT. QED

APPENDIX C:

The original proof of 1965: Close’s Fermat’s Last Theorem (FLT65)

PROOF OF FERMAT’S LAST THEOREM

E. R. CLOSE

The theorem:                 X^N + Y^N ≠ Z^N

When X, Y and Z are integers > 0

And N is any prime integer >2.

The following theorem and corollaries are proved here because their proof makes their application clearer and because certain aspects, not ordinarily mentioned, are of particular interest in the proof of Fermat’s last theorem.

THEOREM: (The division algorithm) If g(X) ≠ 0 and f(X) are any two polynomials over a field, of degree m and n, respectively, and n>m, then there exist unique polynomials q(X) and r(X) such that

(A)        f(X) = q(X)g(X) + r(X)

Where r(X) is either zero or of degree smaller than m.

Let f(X) = a_nXⁿ + a_n-1X ^n-1 +•••+ a₁X+ a₀

And g(X) = b_mX^m + b_m-1X ^m-1 +•••+ b₁X+ b_0, b_m ≠0.

If q(X) is zero or of a degree smaller than m we have

f(X) = 0•g(X) + r(X)

and if n = m, q(X) becomes a constant, Q, and r(X) may be of any degree from zero to n, depending on the value of the constant Q. Thus q(X) and r(X) are not unique in this case.

Now we can form f₁(X), of lower degree than f(X), by writing

(B)        f₁(X) = f(X) - a_n/a_m X^n-m •g(X).

We may now complete the proof of this theorem by induction: Assume the algorithm true for all polynomials over the field. Since f₁(X) is such a polynomial, we may write:

(D)    f(X) = q₁(X)g₁(X) + r₁(X),

where r₁(X) is either zero or of a degree less than m. So from (B.) and (C.):

f(X) = | a_n/a_m X^n-m + q₁(X) | g₁(X) + r₁(X).

or f(X) = Q(X)g(X) + R(X), the desired form of f(X). All that remains is to prove that Q(X) = q(X) and R(X) = r(X), that is, that q(X) and r(X) are unique when m>n.

Now, f(X) = q(X)g(X) + r(X)

And f(X) = Q(X)g(X) + R(X). This implies Q(X)g(X) + R(X) = q(X)g(X) + r(X), or [Q(X) – q(X)]g(X) = r(X) – R(X).

If m = n, the right member of this equation is either zero or of degree less than m, the degree of g(X). Hence, unless Q(X) – q(X) = 0 → Q(X) = q(X) and r(X) – R(X).= 0 → r(X) = R(X)., the left member of the equation will be of a higher degree than m, and we have a contradiction. Thus Q(X) = q(X) and r(X) = R(X), which means that q(X) and r(X) are unique, and the division algorithm is proved.

Again notice the important fact that if m = n, i.e. f(X) and g(X) are of the same degree, Q(X) and q(X) must be of zero degree in X. So R(X) and r(X) are now of the same degree from zero to n, if f(X)X = g(X). The degree of R(X) and r(X) now depend upon the values Q(X) and q(X), so that Q(X) and q(X) are not necessarily equal, as R(X) and r(X)are not, and q(X) and r(X) are not unique.

COROLLARY I: If f(X) and g(X) contain a common factor, r(X) contains it also.

This follows by inspection of equation (A):

            f(X) = q(X)g(X) + r(X) → r(X) = f(X) - q(X)g(X).

COROLLARY II:

The remainder when a polynomial is divided by (X-a) is f(a):.

This follows at once from the division algorithm:

Let g(X) = X-a. Then (A) becomes

            f(X) = (X-a)q(X) + r(X).

By substitution,

f(a) = (a-a)q(a) + r, where r is an element of the field

or, f(a) = r,

so that f(X) = (X-a)q(X) + f(a).

Notice that if q(X) and r(X) are unique, f(a) cannot contain X-a, and so it follows that

COROLLARY III: A polynomial, f(X), of degree greater than one is divisible by X-a IF AND ONLY IF, f(a) = 0.

It may be remarked that the division algorithm and all three corollaries hold when f(X) and g(X) are expressions involving only integers, since integers are elements in the field of real numbers.

THE PROOF

Consider the equation

(1.)     X^N + Y^N = Z^N, where n is a prime number>0, and, X,Y and Z are relatively prime integers.>0. X,Y and Z may be considered to be relatively prime because if two of them, say X and Y, contain a common factor, M, then Z contains it also:

If X = Mx and Y = My, then Z^N = (Mx)^N + (My)^N = M^N(X^N + Y^N). And thus Z = Mz, i.e. Z contains M. Also, X^N + Y^N = Z^N → (Mx)^N + (My)^N = (Mz)^N. Factoring M^N out, we have x^N + y^N = z^N

And so, any case of equation (1.) with X, Y and Z not relatively prime implies a case wherein they are relatively prime, and thus if we prove the theorem for X, Y and Z relatively prime, no non-relatively prime case can exist.

Furthermore, it is only necessary to consider N prime, since any non-prime case for N in (1.) implies a case wherein N is prime. For example, let N = ab, with b prime and a may be another prime or a composite of primes. Then

X^ab + Y^ab = Z^ab → (X^a)^b + (Y^a)^b =(Z^a)^b Clearly a case of (1.) with n prime. Since all non-prime integers are factorable into prime numbers, it is sufficient to consider N as a prime in any proof.

Now, X^N + Y^N = Z^N → Z^N – X^N = Y^N . Factoring, we have:

(2.)     (Z-X)( Z^N-1 + Z^N-2X + Z^N-3X² +•••+ X^N-1) = Y^N .

For any given X, say X = X₁ let Z^N-1 + Z^N-2X₁ + Z^N-3X₁² +•••+ X₁^N-1 = f(Z)

And Z-X₁ = g(Z). Then

(3.)     g(Z)f(Z) = Y^N.

Remembering that for Fermat’s last theorem to be falsified, Y is an integer, as are X and Z, then for any particular case of X^N + Y^N = Z^N, it follows that either g(Z) and f(Z) contain a common factor, or they are both perfect N-powers of integers. By Corollary I, the remainder, when f(Z) is divided by g(Z), will contain any and all factors common to both. And by COR II, the remainder when f(Z) is divided by g(Z) = Z-X₁, will be f(X₁). And

(4.)     f(X₁).= X₁^N-1 + X₁^N-2X₁ + X₁^N-3X₁² +•••+ X₁^N-1 = NX₁^N-1.

Since X, Y and Z are relatively prime, no factor of X₁ may be contained in Z –X. Therefore, if f(Z) and g(Z) have a common factor, it must be N. It also follows that either f(Z) or g(Z) contains N or they are perfect N-powers.

Similarly, from X^N + Y^N = Z^N,

(5.)     (Z-Y)( Z^N-1 + Z^N-2Y + Z^N-3Y² +•••+ Y^N-1) = X^N.

And by exactly the same reasoning as above, for any particular Y = Y₁, if we let

Z^N-1 – Z^N-2Y₁ + Z^N-3Y₁ ² -•••+ Y₁ ^N-1 = f₁(Z) and

Z - Y₁ = g₁(Z), then either f₁(Z) and g₁(Z) contain N as a common factor, or they are perfect N-powers. So for a given case of Z^N = X₁^N + Y₁^N , if either f(Z) or f₁(Z) contains N, the other has to be a perfect N-power, since we have concluded that we only have to consider relatively prime X, Y and Z, implying both cannot contain N.

Now, N ε f(Z) → N ε Y₁ and N ε f₁(Z) → N ε X₁. But X₁.and Y₁ are relatively prime. If neither X₁.nor Y₁ contains N, both, being relatively prime, must be perfect N-powers. Therefore, one of them at least, must be a perfect N-power, not containing N as a factor.

Therefore, f(Z) = Z^N-1 + Z^N-2X₁ + Z^N-3X₁² +•••+ X₁^N-1 = A^N, and/or

f₁(Z) = Z^N-1 – Z^N-2Y₁ + Z^N-3Y₁ ² -•••+ Y₁^N-1 = B^N, A and B integers <Z. and at least one does not contain N.

Since they are both of the same form, set f(Z) = Z^N-1 + Z^N-2X₁ + Z^N-3X₁² +•••+ X₁^N-1 = A^N. Then f(Z) is divisible by A, and since A is a positive integer < Z, we may write A = Z - a, where a is another integer smaller than Z.

The division algorithm tells us that for g(Z) ≠ 0 and f(Z), two polynomials over the field of real numbers, with degrees m and n respectively, and n>m≧ 1, there exist unique polynomials q(Z) and r(Z) such that f(Z) = q(Z)g(Z) + r(Z), where r(Z) is either zero or of degree smaller than m. Notice that if n = m = 1, as in the case when N = 2, or if f(Z) is not equal to an integer raised to the N^th power, as in that case when A is the N^th root of a prime number, q(Z) and r(Z) are not unique and COR. II does not hold. But when N>2, and X, Y and Z are integers, COR. II tells us that if g(Z) = Z - a, a polynomial of degree 1 in Z, q(Z) and r(Z) are unique and the remainder, r(Z) will be of degree < m = 1, i.e., zero in Z, a constant, of the form f(a). Therefore:

f(Z) = (Z-a)q(Z) + f(a) over the integer values of the field of real numbers, and by COR. III, f(Z) is divisible by Z - a IF AND ONLY IF f(a) = 0. The division algorithm and its corollaries apply over the field of real numbers, including integers. Thus when X, Y and Z are integers and N>2,

(6.)      (Z - a) ε f(Z) → f(a) = 0.

But f(a) = a^N-1 + a^N-2X₁ + a^N-3X₁² +•••+ X₁^N-1 = 0 is an impossibility because if Fermat’s last theorem is falsified, X₁ and a, are both positive integers. This implies that f(Z) cannot be a perfect integral N-power. Thus we have reached a complete contradiction by assuming X, Y and Z to be integers, and may state that the equation X^N + Y^N ≠ Z^N has no solutions in positive integers when N is an integer > 2. And so the proof of Fermat’s last theorem is complete.

Edward R. Close December 18, 1965

APPENDIX D

ADDENDUM D to FLT65, November, 2011

For Fermat’s last theorem to be falsified, i.e., for the Fermat equation to have integer solutions, Z and a must be integers and f(Z) must equal (Z –a )^N.

 And we see that f(Z) = (Z –a )^N → f(Z) - (Z –a )^N = 0 , and by substitution of the polynomial f(Z) and expansion of the second term,

Z^N-1 + Z^N-2X₁ + Z^N-3 X₁² +•••+ X₁^N-1– [Z^N - NaZ^N-1 + N(N – 1)/2!a² Z^N-2 +•••+ a^N] = 0

Collecting like power-of-Z terms, we see that the left-hand side of this equation is an N^th degree polynomial in Z. Call it G(Z). This equation is an algebraic expression of the requirements that must be met for Fermat’s last theorem to be falsified.

Equation (1.) may also be considered to be an N^th degree polynomial in Z and can be written as follows:

F(Z) = Z^N – X₁^N - Y₁^N = 0, and thus can be expanded to produce:

F(Z) = (Z-X₁)( Z^N-1 + Z^N-2X₁ + Z^N-3 X₁² +•••+ X₁^N-1) - (Z-X₁)(Z –a )^N = 0.

By inspection of this equation, we see that G(Z) is a polynomial factor of F(Z), and Thus the roots of G(Z) must also be roots of equation (1.).

The fundamental theorem of algebra tells us that any polynomial of degree N has exactly N solutions. Assuming there is a positive integral solution, Z₁ = r_i, (1 ≤ i ≤ N) of the Fermat equation for specific integers X₁ and Y₁, the N solutions (roots) of G(Z), represent all the N possible solutions of equation (1.), (X,Y,Z), Z = r₁, r₂, r₃, … r_N, for the specific values, X = X₁, and Y₁^N = (Z –a)^N f(Z). So any integer factor of f(a) that might represent an integer solution of (1.) must also be a factor of Z – a and G(Z).

In order for any integer factor of f(a) to represent an integral solution (X ₁,Y₁, Z₁) of F(Z), it must be equal to some integer, r, where r is one of the roots, r₁, r₂, r₃, … r_N, of the polynomial F(Z). But, if r is a root of F(Z) = 0, it must also be one of the possible values of the one-degree polynomial Z – a, a polynomial factor of F(Z). If this is true, then by the division algorithm, F(Z)/(Z – a) = Q(Z) + F(a), and by Corollary III, F(a) must equal zero.

The equation F(a) = (a-X₁)( a^N-1 + a^N-2X₁ + a^N-3 X₁² +•••+ X₁^N-1) = 0 is satisfied by:

 a - X₁ = 0 and a^N-1 + a^N-2X₁ + a^N-3 X₁² +•••+ X₁^N-1 = 0

So the N solutions of F(a) are a = X₁ and N – 1 values of a_i, none of which can be integers. The nature of the N – 1 values of a_i, irrational, imaginary or complex, can be determined, but for the purpose of this proof, it is only necessary to observe that they cannot be integers since the sum of N positive integers cannot equal zero.

These N values of a yield the N solutions of F(Z) = 0 as follows:

Z-X₁ = 0 → Y₁ = 0 → X₁^N + (0)^N = Z^N , an integral solution, albeit trivial*, and

Z – a_i = 0, i = 1,2,3,…N – 1 → X₁ ^N + Y₁ ^N = (a_i )^N, where none of the a_i are integers.

 Therefore the N roots, of equation (1.): r₁, r₂, r₃, … r_N, are equal to the N roots of F(Z): X₁, a₁, a₂ , a₃ , … a_in-1.

Since all of the N solutions of the Fermat equation are accounted for, and their numeric nature identified, for N equal to any prime number >2, the proof is complete.

• Note that Fermat, in his statement of the theorem, specified that there could be no positive solutions (X,Y,Z), excluding the solution (X,Y,0). However, X₁^N + (0)^N = Z^N, is, of course, one of the N legitimate solution of F(Z)

REFERENCES

[1]           K.A. Ribet and B. Hayes, “Fermat’s Last Theorem and Modern Arithmetic”, American Scientist 82, March – April, 1994, page 145

[2]           E.R. Close, “The Book of Atma”, Libra Publishers, New York, 1977. Appendix B: 1965 Proof of Fermat’s Last Theorem”,  pp 93-99.

[3]           E.R. Close, “The Book of Atma”,  Libra Publishers, New York, 1977.  Appendix A: Derivation of the “Ratio Formula”,   pp 90-92.

[4]           W. H. Bussey, Fermat's Method of Infinite Descent, The American Mathematical Monthly, Vol. 25, No. 8, pp. 333-337, 1918

[5]        B. F. de Bessy, Traité des Triangles Rectangles en Nombres, vol. I, 1676, Paris. Reprinted in Mém. Acad. Roy. Sci.,5, 1666–1699, 1729

[6]          L. Euler, "Theorematum quorundam arithmeticorum demonstrationes". Comm. Acad. Sci. Petrop. 10: 125–146. Reprinted Opera omnia, ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner, 1915

[7]          A. Legendre, “Théorie des Nombres” (Volume II)(3rd Ed.). Paris: Firmin Didot Frères, 1930. Reprinted in 1955 by A. Blanchard (Paris)

[8]          D. Hilbert, “Die Theorie der algebraischen Zahlkörper".Jahresbericht der Deutschen Mathematiker-Vereinigung 4: 175–546. Reprinted in 1965 in Gesammelte Abhandlungen, vol. I by New York: Chelsea

[9]          L. Kronecker , “Vorlesungen über Zahlentheorie”, vol. I. Leipzig: Teubner pp. 35–38, 1901. Reprinted by New York: Springer-Verlag in 1978

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[1] This is an example of one of the key critiques of FLT65 in its original format:

“F(Z) = Z^N – X₁^N - Y₁^N = 0, and thus can be expanded to produce:

F(Z) = (Z-X₁)( Z^N-1 + Z^N-2X₁ + Z^N-3 X₁² +•••+ X₁^N-1) - (Z-X₁)(Z –a )^N = 0.integers

By inspection of this equation, we see that G(Z) is a polynomial factor of F(Z) ... We can see the error here by using z for the variable and Z for the constant figuring in the Fermat eqn….”

In this “clarified” version (FLT65C), we will see that this problem never actually arises because z of the FLT equation is always a variable, either an unrestricted variable, z, or an integer variable, Z, never a constant, throughout the proof. The fact that this is not clearly indicated in the notation of the original FLT65 proof has caused confusion of the sort expressed in the example critique above. This confusion is fully clarified in this presentation. The key to understanding the proof is realizing that the legitimate application of the division algorithm and corollaries to the polynomial factors of the FLT equation, as polynomials in the variable z, leads to a unique remainder which reveals the fact that if x and y are integers, z cannot be an integer. Thus the question of factors of polynomials in the variable z, versus constant integer factors is resolved. Other than the need for clearer notation, this question is the only legitimate concern that reviewers of FLT65 have brought to my attention.

[2] In mathematics, a field is a group of ordered quantitative elements that contains a multiplicative inverse for every nonzero element. As such it is an arithmetical structure within which the operations of addition, subtraction, multiplication, and division are consistent and valid, and it supports an algebraic structure within which the same basic operations are valid. Geometrically, a field is a dimensional domain within which arithmetically and algebraically defined elements can be described and located. The three most basic mathematical fields are the field of real numbers, the field of rational numbers, and the field of complex numbers.

[3] FLT65C is the 2014 rewrite of FLT65 but without modifying any of the Equation statements. The proof above is FLT65C.

[4] A ring is a set, S, of mathematical or algebraic elements for which the four basic operations of addition, subtraction, multiplication, and division apply. And if a, b and c represent elements of a ring, the four basic operations satisfy the following conditions:

1. Members of the set are additively associative: For all a, b and c, (a + b) + c = a + (b + c)

2. They are additively and multiplicatively commutative: For all a and b, a + b = b + a,, and a x b is equal to b x a.

3. There exists a zero element, or additive identity, such that for all a, 0 + a = a+ 0 = a.

4. There exists an additive inverse: For every a there exists – a, such that a + (-a) = (-a) + = 0.

5. Added elements are multiplicatively distributive: For all a, b and c, ax(b + c) = axb + .axc and (b + c)xa = bxa + cxa.

6. Elements are multiplicatively associative: For all a, b and c, (a xb)xc = ax(bxc).

The simplest example of a ring is the set of integers, and the set of integer polynomials also forms a ring. The field of real numbers is also a ring, but, even though the ring of integers is a subset of the field of real numbers, it is not a field because its elements do not have multiplicative inverses.

[5] Infinite descent is a powerful method for proving or disproving propositions involving integers. In general, the method, which appears to have been one of Pierre de Fermat’s favorite methods of proof, may be described as follows: if ᵱ is a property that integers or functions of integers may possess, and if the assumption that a given positive integer, N, or a function based on it has the property ᵱ leads by a mathematical process of one or more steps to the existence of a smaller positive integer, N₁ < N, that also has or provides a function that has the property ᵱ, then no positive integer or form of the function involved can have that property. This conclusion is logically and mathematically valid because repeated applications of the same process that led from N to N₁, will produce a series of integers: N > N₁ > N₂ >…> N_i,. that also have the property. Since the process can be repeated again and again, leading to an infinitely decreasing sequence of positive integers - which is impossible - the assumption that ᵱ is possessed by a given positive integer implies a contradiction and, hence, is false. This method may be applied to a set of integers, sums of integers, and any function that is reducible to an integer.

The method of infinite descent is commonly associated with the French mathematician Pierre de Fermat, probably because he was the first to state it explicitly.^[4]

[6] Around 1650, Pierre de Fermat used the technique of infinite descent to prove that the area of a right triangle with integer sides can never equal the square of an integer.^[4] Expressing this geometric proposition algebraically with the equation x⁴ + y⁴ = z², he proved the proposition by proving that this equation has no relatively prime integer solutions. He noted that this also proves what we now know as FLT for the case n = 4, since the equation a⁴ + b⁴ =c⁴ can be written as c⁴ − b⁴ = (a²)². Later, a number of professional mathematicians produced proofs for the case n = 4, including Bernard de Bessy (1676),^[5] Leonhard Euler (1738)^[6], Adrien-Marie Legendre (1830),^[7] David Hilbert (1897),^[8] and Leopold Kronecker (1901),^[9].