FLT65 PROOF PART 5

FERMAT'S LAST THEOREM PART 5

FERMAT’S LAST THEOREM AND THE CLOSE FLT65 PROOF

Before explaining in detail how some reviewers have missed certain critical details of FLT65, and demonstrating why their concerns are unwarranted, and why their ‘counterexamples’ are invalid, a brief description of FLT and FLT65 is in order.

Fermat’s Last Theorem (FLT) states that xⁿ + yⁿ ≠ zⁿ, when n is an integer greater than 2 and x, y, and z are integers. FLT is stated here in the negative, as an inequality (“it cannot equal”), to emphasize the fact that the FLT65 proof consists of the negation of the existence of the equation xⁿ + yⁿ = zⁿ when x, y and z are integers with n >2. (The cases n = 1 and n = 2 were known to have infinitely many solutions and are thus not part of FLT.)

FLT65 is based on, and relies upon application of the division algorithm to algebraic polynomials of real variables, and in the case of FLT, to integer polynomials because only integer solutions to the FLT equation are of interest. This reliance is of such basic importance that complete proofs of the algorithm and three corollaries, with requirements for their application were included in the 1965 proof to help clarify the proof of FLT. Reviewers who skipped over the division algorithm proof in FLT65, might have missed the importance of the uniqueness requirements which are very relevant to the division algorithm as applied to the FLT equation. It is important to note that they are clearly stated and their importance is explained in detail in the first three pages of FLT65. See Appendix A.

Brief statements of the division algorithm and its three relevant corollaries, with examples to show how they apply, are provided below. For convenience and quick reference, proofs of the division algorithm and its corollaries are also provided separately in Appendix B of this presentation.

The examples the division algorithm and corollaries provided below are specifically designed to show the relevance to FLT65.

FLT65 focuses on n = prime numbers greater than 2 because non-prime exponents can always be factored into prime numbers and the case of n = 4 had already been proved by many. It is, therefore, clear that it is sufficient for FLT65 to focus purely on n = prime numbers greater than 2. FLT65 proves FLT by demonstrating that assuming an integer solution of the FLT equation produces a contradictory inequality when n is prime greater than 2.

See Appendix C for formal proof that it is sufficient to prove FLT for n equal to prime numbers greater than 2.

The Division Algorithm

The division algorithm is an algebraic expression of the basic relationship between the

·       dividend, f,

·       divisor, g,

·       quotient, q, and

·       remainder, r,

in the process of division, where f(X), g(X), q(X), and r(X) are algebraic functions of some variable, represented here by the symbol X.

The division algorithm is stated as follows: For any two polynomials, g(X) ≠ 0 and f(X) over the field of real numbers, of degree m and n, respectively, and n > m > 0, there exist unique polynomials such that f(X) = q(X)g(X) + r(X) where r(X) is either zero or of degree smaller than m.  

The Division Algorithm is true for any real value of X, rational, irrational or integer.

(Note that in  the Division Algorithm, n can be any positive integer, 1, 2, 3,… , while, in the FLT65 proof, n is restricted to integer values > 2.)

Example: Let f(X) = X⁴ + AX³ + A²X² + A³ X + A⁴, and g(X) =X - a. Then, for this polynomial (a polynomial of the same form as f(Z) in FLT65)the mathematical operation of division, represented by f/g = q + r, becomes:

(X⁴ + AX³ + A²X² + A³ X + A⁴)/(X-a) = q(X) + r(X)/(X-a), where, by the division algorithm, r(X) is either zero or of degree less than 4. Carrying out the polynomial long division to determine the algebraic polynomial form of q and r, we get:

q(X)= X³ + (A+a)X² +(A² +aA+a²)X+(A³  +aA² + a²A² + a³) and r(X)= A⁴  +aA³ + a²A² +  a³A +a⁴. And if X and a are positive, r(X) cannot be zero and is not of degree smaller than m, so by the division algorithm stated above, the polynomial (X⁴ + AX³ + A²X² + A³ X + A⁴) is not divisible by the polynomial (X-a). But what about the integer values of these algebraic polynomials?

We can assign A, X and a integer values to illustrate the transition from algebraic polynomials to integers: Let A = 1, a = 2, and X = 5, and substitute these integer values into f(X) = q(X)(X-a) + r(X). This gives us:

 (5⁴ + 5³ + 5² + 5 + 1)= (5³ + 3x5² +7x5+15)(5-2)+31→ 781 = 250x3+31 = 781.

The values of the integer polynomials in this example, reduced to single integers, are f(X)=781, the divisor, X-a =3, q(X)=250, and r(X)=31.

We see that the integer value of r(X) is 31, which is not divisible by the integer value of X-a = 3. This example demonstrates that f(X), which is of the same form as f(Z) in FLT65 with n = 5, is not divisible by X-a, of the same form as Z-a, and that this is true for both the algebraic polynomials, and for  their reduced integer values.

This example illustrates the relationship between the polynomial remainder and integer remainder when the f(Z) factor of the FLT equation is divided by Z -a, which is especially important in the application of  corollary III of the division algorithm in FLT65 and as discussed below.

Corollary I: If f(X) and g(X) contain a common factor, r(X) contains it also. It is important to note that this corollary is true for algebraic polynomials over the field of real numbers, integer polynomials and the integer values of those integer polynomials.

If f(X) and g(X) are integer polynomials, i.e., all terms and coefficients are integers and X is an integer variable, and they contain a common factor, X-a, then the remainder integer variable polynomial r(X) contains X-a, and the integer value of the remainder also contains the integer value of X-a.

Example: the polynomials f(X) = X⁴+4X³-X² -16X-12 and g(X) = X²-4 contain the common factor X–2. Corollary II says that because f(X) = X⁴+4X³-X² -16X-12 = q(X)(X²-2) + r(X), r(X) contains X – 2, and we  can see that this is true for both algebraic polynomials and their integer values, as long as X-2 is not equal to zero.

For polynomials with X variable, f(X) = (X+1)(X+2)(X +3)(X-2) = q(X)(X+2)(X-2) + r(X), which gives us r(X)=(X+1)(X+2)(X +3)(X-2)-q(X)(X+2)(X-2)= {(X+1)(X+2)(X +3)-q(X)(X+2}(X-2), so we can see by simple inspection of this equation that r(X) contains X-2.

Since the divisor cannot equal zero, X must be greater than 2. So, for this demonstration, let X = 5, then X – a = 5 – 2 = 3, and factoring f and g, we have

(X+1)(X+2)(X +3)(X-2)=q(X)(X+2)(X-2)+r. substituting X=5.this becomes

6x7x8x3 =7x3xq + r, and even without going farther, we can see by inspecting this equation that r contains 3, the integer value of X-2.

Carrying out the algebraic and substituting X = 5, we get f(X) = X⁴+4X³-X² -16X-12q(X) = 1008, q(X) = X²+4X+3 = 882, and r = 6X²- 24 = 150 – 24 = 126 = 3x42.

We can check our results for r by substituting the integer value of the integer polynomials f, q and g into f(X) = q(X)g(Z) + r(X): 1008 = 882x21 + r → r = 1008 – 882 = 126.

This shows r = 126 = 3x42, verifying the fact that r contains the integer value of X–a.

Corollary II: The remainder when a polynomial f(X) is divided by (X–a) is f(a).

Division Algorithm Corollary III: Given a polynomial, f(X), of degree greater than one, if q and r are unique, then f(X) is divisible by (X – a) IF, AND ONLY IF, f(a) = 0.

Using the example presented for the division algorithm above:

 f(X) = X⁴ + AX³ + A²X² + A³ X + A⁴, A =1 and a = 2, so that f(X) = X⁴ + X³ + X² + X + 1.

Evaluating f(a) for this example, f(a) = 2⁴ + 2³ + 2² + 2 + 1=31, which is clearly non-zero, confirming what we found by polynomial long division and integer substitution above.

Using the example for corollary I, presented above, f(X) = X⁴+4X³-X² -16X-12 and a = 2. Therefore, f(a) = 2⁴+4x2³-2² -16x2-12 = 16+32-4-32-12 = 0, confirming that f(X) is divisible by X–2, which we found above by factoring f(X) and substituting integer values into the algebraic polynomial forms of the division algorithm equation.  

_______

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, for p a prime number and for X₁, Y₁ and Z₁ relatively prime.

Proofs that these two conditions, namely

·       It is necessary and sufficient to prove FLT for the exponent n equal to primes greater than 3 (n = p ≥ 3), and

·       It is necessary and sufficient to prove that  there is no solution (X₁, Y₁, Z₁) such that X₁, Y₁, and Z₁ are relatively prime integers

Proofs that these conditions are necessary and sufficient are given in FLT65. For efficiency, they are not included here, but for easy reference, completeness and clarity, they are presented in Appendix C.

Appendices

APPENDIX A:

The original published proof of FLT65: Close’s Fermat’s Last Theorem

Including proof of the Division Algorithm and its corollaries

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:

(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.

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) = 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 1 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 B ^[18]

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 C:

PROOF OF SUFFICIENCY FOR X, Y AND Z RELATIVELY PRIME

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

Proofs of the sufficiency of considering X,Y and Z relatively prime, and N restricted to primes greater than 2 were included in FLT65 as published in the Book of Atma in 1977. The author developed the proof of FLT for N = 4 while finishing his degree in mathematics in 1962, but never published it because it had already been published by others. In retrospect, it may have been worthy of publication because of its utter simplicity. It is a simple, straight-forward extension of the Derivation of the Ratio Formula as published in Appendix A of the Book of Atma.

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). This proof was included in FLT65. N= 4 is a special case since 4 is neither prime nor a multiple of primes > 2.

PROOF OF FLT FOR N = 4 BY INFINITE DESCENT

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, the answer is no, because we can eliminate all even powers of 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: Derivation of the Pythagorean triples formula from properties of rational numbers was published by the author in 1977^[3].

Using the formulae derived for the Pythagorean triples, 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, and
Z² = P² + Q²

Now with this application of the Pythagorean triple formulae, we have obtained another Pythagorean triple: Y² + Q² = P². By comparison with the equation (X²)² + (Y²)² = (Z²)² and the formulae above, we see that P < Z², Q< X², and Y < Y².

Thus, by assuming that a triple integer non-zero solution exists for the equation (X²)² + (Y²)² = (Z²)² → (2PQ)² + (P² - Q²)² = (P² + Q²)² we can produce another integer triple solution with smaller integers: Y² + Q² = P².

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 two 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 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 other non-prime odd integers are factorable into prime numbers, it is sufficient to consider N = p, a prime in any proof of FLT. QED.

FLT65: FERMAT'S LAST THEOREM PART 4

FERMAT'S LAST THEOREM PART 4

NOTE: This part contains some original mathematics. The theorems stated and proved in this section are used in the explanation that follows, but If math is not your thing, you can read the text to get the gist of it and skip over the math. The references cited will be posted with the last part of this paper.

A NOTE ON THE NATURE OF MATHEMATICAL LOGIC AND PROOF

If FLT65 is a valid proof of Fermat’s Last Theorem, why hasn’t it been recognized in all these years? There are a number of reasons, and they are discussed below, but an important underlying reason is the lack of consensus among reviewers concerning exactly what constitutes mathematical logic and proof. We must, therefore, start by explaining what is meant by those terms in this presentation:

Mathematical logic starts with

(1.) the inductive reasoning process of identifying self-evident axioms underlying the logical patterns of reality, like distinctions of subject and object, separation and unity, enumeration and combination, difference and equivalence, extension and content, dependent and independent variables, functional relationships, and

(2.) formalizing the fundamental operations of substitution, addition, subtraction, multiplication and division. Based on such axiomatic concepts and fundamental operations, the logical structures of mathematical systems are built up by processes of deductive reasoning.

 (3.) Mathematical proof consists of using mathematical operations and substitutions of equivalents to determine whether a given statement or proposed theorem is consistent with accepted mathematical logic and the axioms underlying that system.

This sounds like a boring, rigidly structured affair that might be completed in short order. But that is, in fact, not the case.

The history of FLT65 provides a good example of how mathematics is not the ‘cut and dried’ subject that most people seem to think it is. One reviewer of FLT65, a highly intelligent man with deep interests in science and mathematics and significant math skills— but not a professional mathematician —said, after discussing FLT65 for some time with  the author, something to the effect that, “of all things, a mathematical statement like a proposed proof of FLT is something we should absolutely be able to come to an agreement about in short order, because it’s either true or it’s not.” This reflects an erroneous belief about mathematics that was common, even among professional mathematicians, including the great German mathematician David Hilbert, before Kurt Gödel published his proofs of two theorems now known as Gӧdel’s incompleteness theorems (GIT) (1929 – 1931) ^[6]. Gӧdel’s incompleteness theorems are very relevant to FLT and FLT65 because they explain, in part, why some reviewers have not understood FLT65. Additionally, after Gӧdel’s work became widely known, many mathematicians suspected that FLT might actually be an unprovable statement.

Gӧdel’s work was a bombshell that changed mathematics forever: It destroyed Hilbert’s dream of establishing all mathematical logic upon a finite, consistent foundation of axioms, from which all statements could be proved either true, false or meaningless by deductive reasoning. Finding such an all-encompassing algorithmic system was known as the ‘entscheidungsproblem’ (decision problem) ^[7]. Gödel proved, however, that no system of mathematical logic is ever complete. This means that there can be no complete entscheidung algorithm for all mathematical statements. It means that there can be logically legitimate questions posed within any given axiomatic mathematical system that are meaningful, but unprovable within that system.

In the introduction to his breakthrough paper, Gödel said: “As is well known, the development of mathematics towards greater certainty has led to the formalization of much of it such that you can complete proofs by mechanically following a few basic rules. The most comprehensive current formal axiomatic systems are the system of the Whitehead-Russell Principia Mathematica on the one hand, and the Zermelo-Fraenkel axiom-system of set theory on the other hand. These two systems are so thoroughly developed that one can formalize in them all methods of proof currently in use in mathematics, i.e. you can reduce the methods of constructing proofs to a few axioms and deduction rules. Therefore, it seems plausible to [as Hilbert did][1] that these deduction rules are sufficient to decide all mathematical questions expressible in those systems. We will show that this is not true, but that there are even relatively easy problems in the theory of ordinary whole numbers that cannot be decided from existing axioms.”^[8]

 A mathematical statement that has not been proved is properly called a conjecture. With a conjecture, the question to be decided is whether the conjecture is true or not. After Gödel, some mathematicians believed that Fermat’s conjecture (FLT), a legitimate mathematical question, might be an unprovable statement. This idea, however, was based on a misunderstanding of the incompleteness theorems. The incompleteness theorems do not, as one might suppose, imply that questions that cannot be decided within the axiomatic system they are framed in, are necessarily ultimately unprovable. Gӧdel’s theorems did not prove that there are undecidable questions, they proved that there are some questions for which the mathematical system within which one is operating may be inadequate. The system of mathematical logic within which a conjecture is stated may have to be expanded with one or more new axioms and new definitions and theorems before the statement can be proved to be either true or false.

In order for a reader to understand the applicability of the “Division Algorithm” and its three corollaries as they are used in FLT65, it may be necessary to expand the logic of the mathematical system from which the reviewer is operating to include the following definitions, axiomatic principle and theorems:

Concepts of importance in FLT65

1A   Mutually Co-Dependent Variables.

Variables are mutually co-dependent if they are related only by the form of the algebraic expression or equation they appear in, with no designated difference between them in terms of dependence or independence. For example, in the equation zⁿ - xⁿ = yⁿ, the variables x, y and z are mutually co-dependent. If we designate one of them as the dependent variable and assign values to the others so that we can set up an equation in terms of functions of the dependent variable, the variables are no longer mutually co-dependent. For example, if x and y are replaced by specific types of variables or constants, like X and Y for integer variables, or specific values, X₁ and Y₁, z becomes dependent upon their values and we say the equation is of the nth degree in z.[2]

1B. The Principle of Designation of Dependent and Independent Variables:

Given a polynomial f(x, y, z, … ) a function of several variables, over the field of real numbers, any of the otherwise mutually co-dependent variables, x, y, z, …, may be designated as dependent and/or independent, as appropriate to make the polynomial representative of meaningful information as part of an equation or descriptive statement.

This statement expresses an inherent fundamental feature of algebraic expressions consistent with the axioms of a system of mathematical logic. It is a principle rather than a theorem because, while it depends upon the validity of the axioms and fundamental operations of mathematics, it requires no multi-step proof by deductive reasoning from those axioms. This operational principle allows us to use polynomials to set up potentially solvable equations for application to real problems. This principle is relevant to FLT65 and this discussion. This is so because at several points, the designations of dependent and independent variables are changed as needed to express functional relationships consistent with application of the division algorithm. It was used without comment in FLT65, and its use is made explicit here for more clarity.

2A. Parallel Polynomials:

Parallel polynomials are polynomials of the same degree and the same algebraic form, i.e. with the same number of terms and the same coefficients, but with different variables. For example, a₁xⁿ + a₂x^n-1 + a₃x^n-2 + … + a_n+1 and a₁z ⁿ + a₂z^n-1 + a₃z^n-2 + … + a_n+1 are parallel polynomials, as are Az² + Bzx + Cx² and Ax² + Byx + Cy².

2B. The Theorem of Parallel Factorization:

If a given polynomial, f(z), of degree n > 1, defined over the field of real numbers, is divisible by z –a, then the parallel polynomial f(a) is also divisible by z – a.

This theorem is proved indirectly in FLT65, and is proved formally here for the first time, as far as the author knows. It could also be considered to be a corollary to the division algorithm. Note that the choice of symbols used to represent variables in proofs is arbitrary: The validity of axioms and theorems do not depend upon the symbols used—the use of any consistent set of symbols would produce the same theorems. However, the symbols a, x, y, and z were chosen to correspond to the symbols used throughout this paper to facilitate understanding and application of the theorems to FLT65. In the generalized two-variable algebraic polynomial forms used in the following proofs, x and z were chosen for the same reason.

Proof: Consider the generalized n-degree polynomial:

f(z, x) =Azⁿ +Bz^n-1x +Cz^n-2x² +…+Mxⁿ. …………………………………………………………..       1

By the Axiom of Unrestricted Designation of Variables, we may designate x = X₁, to obtain the single-variable polynomial

f(z) =Azⁿ +Bz^n-1X₁ +Cz^n-2X₁² + … + MX₁ⁿ  ……………………………………………………….  2

If we divide f(z) by z – a, by Corollary II of the Division Algorithm, the remainder,

f(a) = Aaⁿ +Ba^n-1X₁ +Ca^n-2X₁² +…+ MX₁ⁿ. ……………………………………………….  3

Then, f(z) – f(a) = Azⁿ - Aaⁿ + Bz^n-1X₁ - Ba^n-1X₁² + Cz^n-2X₁ ² - Ca^n-2X₁² + …+ LzX₁^n-1 - LaX₁^n-1 + Mxⁿ - MX₁ⁿ = A(zⁿ - aⁿ) + B X₁(z^n-1 - aⁿ) +… + 0 = A(z – a)(z^n-1 + z^n-2a +…+ a^n-1) +BX₁(z – a)(z^n-2 + z^n-3a +…+ a^n-2) +…+ LX₁^n-1(z - a) →f(z) – f(a) = (z – a) {A(z^n-1 + z^n-2a + … + a^n-1) + B X₁(z^n-2 + z^n-3a + … + a^n-2) +…+ LX₁^n-1}.  ………………………………….. 4

Then, since f(z) is divisible by z – a, and f(z) – f(a) = (z – a) {A(z^n-1 + … + a^n-1)+B X₁(z^n-2 + … + a^n-2) +…+ L X₁^n-1}, by Corollary I of the Division Algorithm, f(a) is also divisible by z – a, Q.E.D. ………………………………………………………………………………  5

Use of this theorem shortens the discussion of FLT and FLT65 considerably, as the reader will see in the discussions that follow. Also, proof of the companion theorem for integers given below is important because it removes some of the mystery expressed by reviewers concerning the application of theorems that are true for algebraic polynomials to the subgroup involving integers, namely integer polynomials.

2C. The Theorem of Parallel Factorization for Polynomials of Integer Variables:

 If a given polynomial, f(Z), defined over the ring of integers is divisible by Z –a, the parallel polynomial f(a) defined over the ring of integers is also divisible by Z – a.

This can be inferred from the proof of the theorem of parallel factorization for polynomials over the field on real numbers and the fact that the integers are a subset of the set of real numbers. Or, stated more formally: It follows from the fact that all of the elements of the ring of integers are contained within the set of elements of the field of real numbers. However, importantly, it can be proved independent of the proof for polynomials over the field of real numbers as follows:

Proof:

Given the general polynomial in two variables:

f(z,x) =Azⁿ +Bz^n-1x +Cz^n-2x² +…+Mxⁿ . ………………………………………………. 1

By the Axiom of Unrestricted Designation of Variables, we may designate x = X₁ a specific integer and z = Z, an integer variable, in the general polynomial to obtain f(Z) =AZⁿ +BZ^n-1X₁ +CZ^n-2X₁² +…+MX₁ⁿ. ……………………………………………. 2

And then, when we divide f(Z) by Z – a, by Corollary II of the Division Algorithm, the remainder is the parallel polynomial

f(a) = Aaⁿ +Ba^n-1X₁ +Ca^n-2X₁² +…+MX₁ⁿ. ……………………………………………… 3

 Then, f(Z)–f(a) = AZⁿ - Aaⁿ + BZ^n-1X₁ - Ba^n-1X₁² + CZ^n-2X₁ ² - Ca^n-2X₁² + …+ LZX₁^n-1 - LaX₁^n-1 +Mxⁿ - Mxⁿ =A(Zⁿ - aⁿ) + Bx(Z^n-1- aⁿ) +…+ 0 =A(Z – a)(Z^n-1 +Z^n-2a + …+ a^n-1) +BX₁(Z – a)(Z^n-2 +Z^n-3a +…+ a^n-2) +…+LX₁^n-1(Z - a) → f(Z) – f(a) = (Z – a){A(Z^n-1 + Z^n-2a + … + a^n-1) +BX₁(Z^n-2 + Z^n-3a + … + a^n-2) +…+ LZ^n-1} ………………………………………… 4

Then, If f(Z) is divisible by Z – a, and f(Z) – f(a) = (Z – a){A(Z^n-1 +  … + a^n-1) + BX₁(z^n-2 + … + a^n-2) +…+ LX₁^n-1}, then, by Corollary I of the Division Algorithm, f(a) is also divisible by Z – a, Q.E.D. …………………………………………..………………………………… 5

By incorporating the operational principle, definitions and theorems presented above into the axiomatic system of mathematical logic and applying them to FLT, we will show how  the FLT65 demonstration that f(a), the remainder obtained by dividing f(Z) by Z – a can never equal zero [i.e. f(a) ≠ 0], provides an unavoidable contradiction proving Fermat’s Last Theorem.

---

[1] A comment added by the author of this paper.

[2] See Appendix D for detailed notational definitions.

                                    

FERMAT'S LAST THEOREM PART 3

FERMAT'S LAST THEOREM PART 3
Why is FLT65 important in regard to understanding quantum physics and consciousness? Because all reality exists in whole number multiples of the smallest possible quantum. This means that Diophantine (whole number) equations provide the perfect mathematics to describe objective reality, and the equation of Fermat's Last Theorem is a Diophantine equation. When FLT is applied to the Diophantine equations describing the combination of elementary particles, it explains why quarks can only combine in threes. This leads directly to the discovery of the third form of  the substance of reality, linking consciousness to subatomic reality.

A BRIEF HISTORY OF FLT and FLT65.

Sometime in 1636 or 1637, Pierre de Fermat wrote in Latin in the margin of a book on Diophantine equations ^[1] the first known statement of a theorem that became known as “Fermat’s Last Theorem” (FLT). His ‘marvelous’ proof ^[1], was never found and FLT remained without formal proof for more than three centuries. Because of this, it is perhaps the most famous theorem in the history of mathematics. In modern representation, FLT is stated as follows:

“No three positive integers X₁, Y₁, and Z₁ can satisfy the equation x^N +y^N = z^N for any integer value of N >2.” A modern proof, attributed to Sir Andrew Wiles was published 1995. ^[3] This proof was, as indicated, the first generally accepted proof. Dr. Wiles is said to have spent seven years completing it.

In 1965, three decades before Wiles’ proof was announced, the author, Edward R. Close, produced a proof of FLT. It was completed in December 1965 and is, therefore, referred to in this paper as FLT65. A brief summary of the history of the proof is provided, and an exact copy of the original FLT65 proof is presented in Appendix A.

At the time he produced the proof, the author was still early in his career: At that point he held a Bachelor’s degree in mathematics, and was a member of Kappa Mu Epsilon, the National Honorary Mathematics Society.

FLT65 was submitted to a professional mathematician for review for the first time in 1966^[2] and it was first published it in 1977 in a book, as an appendix, pages 93 – 99, in “The Book of Atma” ^[5]. Over the years, the author attempted to get FLT65 peer reviewed and published in mathematics journals several times. These attempts did not succeed for reasons that are discussed in detail in this paper.

The nearest the author came to succeeding was with the Journal of Number Theory in April 1985. The editor then, Dr. Hans Zassenhaus, was encouraging; and, because he could not find a willing peer reviewer, offered to review it himself.  While Dr. Zassenhaus was reviewing FLT65, the author’s career took him to several remote locations in the Middle East over a period of several years, making correspondence very slow and difficult. Before the process could be completed, Dr. Zassenhaus retired, and the next Editor of the Journal was not very interested in “simple” proofs of FLT. Later, upon returning to the US, the author attempted to resume their correspondence and learned that, unfortunately, Dr. Zassenhaus had passed away.

Over the years, the proof has been submitted to more than fifty mathematicians, the great majority of whom found nothing wrong, and no one has actually disproved FLT65.

---

[1] A Diophantine equation is an equation in which only integer solutions are allowed. The infamous difficulty of proving Fermat’s Last Theorem prompted David Hilbert, recognized as one of the most prolific mathematicians of the 19th and early 20th centuries, to include, as number 10 in his List of the most important problems of mathematics, ^[4] the problem of finding a general algorithm for solving Diophantine equations. No such general algorithm has been found to date.

FERMAT'S LAST THEOREM PART 2

PART 2

Fermat's Last Theorem is important in the development of the TRUE unit analysis of quantum data.

“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]

Fermat’s “marvelous” proof was never found and the theorem remained officially a conjecture without proof until Andrew Wiles published a lengthy treatment in 1995 that was accepted by number theorists as a valid proof ^[3]. Prior to that, however, Edward R. Close completed a proof in 1965 (FLT65), submitted it to the first of many reviewers in 1966, and published it in 1977 ^[4]. The Close proof though never refuted, presented difficulties for some reviewers because of unconventional notation, and at least three reviewers have suggested that the difference between applying the division algorithm to algebraic polynomial factors and integer factors of the equation as used in the proof make it questionable and either incomplete or incorrect. This concern is addressed in this paper and shown to be unwarranted: There has been a tendency for reviewers to ignore the uniqueness requirements for the division algorithm stated in FLT65 while looking for mistakes which they assume must be there, for reasons explained in this paper. When everything presented in FLT65 is appropriately considered, the concern over the application of the division algorithm is eliminated, removing the only serious objections to the proof. 

News about my1965 Proof of Fermat's Last Theorem

More than 50 years ago I made some profound discoveries that I’ve used over the years to explore the world of mathematical physics and the relation between quantum mechanics and consciousness. This adventure began when I developed a proof for the famous mathematical conundrum known as Fermat’s Last Theorem in 1965. 
My proof, known as FLT65, has never been refuted. Fermat’s Last Theorem, thought by most to be an abstract number theory theorem, has serious application to quantum physics and It is being currently being reviewed in connection with new research.

This is the first in a series of posts of sections of this paper currently being reviewed by  number of mathematicians and scientists. 

CLARIFICATION AND EXPLANATION OF FLT65

A 1965 PROOF OF FERMAT’S LAST THEOREM

By Edward R. Close, PhD

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. Fermat penned the following statement in Latin in the margin of a book on Diophantine equations in 1637:

“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]

Fermat’s “marvelous” proof was never found and the theorem remained officially a conjecture without proof until Andrew Wiles published a lengthy treatment in 1995 that was accepted by number theorists as a valid proof ^[3]. Prior to that, however, Edward R. Close completed a proof in 1965 (FLT65), submitted it to the first of many reviewers in 1966, and published it in 1977 ^[4]. The Close proof though never refuted, presented difficulties for some reviewers because of unconventional notation, and at least three reviewers have suggested that the difference between applying the division algorithm to algebraic polynomial factors and integer factors of the equation as used in the proof make it questionable and either incomplete or incorrect. This concern is addressed in this paper and shown to be unwarranted: There has been a tendency for reviewers to ignore the uniqueness requirements for the division algorithm stated in FLT65 while looking for mistakes which they assume must be there, for reasons explained in this paper. When everything presented in FLT65 is appropriately considered, the concern over the application of the division algorithm is eliminated, removing the only serious objections to the proof.

______________

In this paper, the author provides a brief description and history of FLT65, followed by presentation and refutation of the objections and supposed counterexamples offered by some reviewers in efforts to invalidate FLT65. Several appendices are attached, including a copy of the original FLT65 proof. The other appendices contain supporting arguments and information confirming that FLT65 was and is, a valid proof, justifying the statement found on the last page of FLT65:

“… 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 ^[2]

GOD, LOVE AND PURPOSE

Moderation in all things is a good rule of thumb, except when falling in love, or seeking to know the true nature of Reality. To know real love is to ascend to the point of being able to touch the ecstatic soul of God, and to know Reality at its deepest level is to know the mind of God; and half-way measures in seeking God/Reality will not suffice.
You cannot stop short of the goal, because a mind with a little knowledge of reality, mixed with either too much fantasy or too much cynicism is a very dangerous thing, And both, as history has shown us, have caused a plethora of suffering and misery, and way too much death and destruction.
The world today is divided as never before, between those on one extreme, like Richard Dawkins and many main-stream scientists, who say there is no need for the concept of a god, and those on the other extreme, like religious zealots who say that if you don’t accept the specific dogma and doctrines of their particular belief system, you are bound for Hell. Most people fall somewhere between these extremes. But this can be the land of Limbo, where there are no goals beyond surviving from day-to-day, no direction, and in such limbo, life has no meaning. This leads to the dark night of the soul and endless despair. So we must avoid the extremes, but live as if your hair were on fire, because it literally is. Your body is designed to withstand the ravages of time and physical entropy for only so long. Your physical body is not likely to last forever, but your essence is spirit, which cannot be burnt by fire, drowned by water, or harmed in any way by the trials and vicissitudes of this world.
Some scientists say that science tells us that there is no God. This is totally false. Science does not tell us there is no God. On the contrary, since the discoveries of the relativistic and quantum nature of physical reality, science has been telling us over and over again that there is more to reality than matter and energy interacting in space and time. Both Planck, the father of quantum physics, and Einstein, the father of relativity, stated in no uncertain terms that there is an intelligence far beyond the mind if man behind it all. You can find their words to this effect in other posts on this blog. 

In fact, the universe without a Supreme Intelligence organizing and sustaining it would rapidly decay to maximum entropy (nothingness). In fact, it is easy to show scientifically and mathematically that there could be no physical reality in the first place without a pervasive creative force. This what my science and Spirituality posts are all about: You need to know that science and spirituality are not mutually exclusive. No one need despair! We are all sons and daughters of the most high God, we were created for the purpose of expressing His Perfection, each in his or her own unique way, in this physical universe.
All goals pale in comparison with the goal of knowing Reality, so we must act as if we might die tomorrow, because until we truly know that we are Spirit, ever in touch with God, we have not achieved the real purpose for which we were born.

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Creation vs Evolution: a Debate Arising from a Deep Misunderstanding

cREATION OR EVOLUTION?

Richard Dawkins, high-profile self-proclaimed atheist is on national TV today making declarative statements like "Evolution is a fact with thousands of proven examples, implying that there was and is no 'Creator'." Mr. Dawkins is either an idiot, or he is being deliberately misleading. Yes things, including biological life, evolve over time, but this in no way eliminates the fact that there has to be an intelligent creative force. In fact, without an intelligent creative force there could be no physical universe. This is true for a number of reasons, not the least of which is the second law of thermodynamics. It is understandable that many may think that the debate over the existence of God vs a universe evolving from the random combination of matter and energy after a big-bang explosion, but this is a misguided debate arising from a profound misunderstanding on the part of the participants on both sides. While there is ample evidence that things evolve, there is also ample evidence of an infinite form of consciousness behind it all. So we need to avoid the extremists' rantings on both sides, and examine the evidence without prejudice. 

The Illusion of Material Reality

Clues from relativity and quantum physics suggest that the time-honored idea that matter, energy, space, and time exist separately is incorrect.  It appears that the macro forms of matter, space and time we perceive through our physical senses are subtle illusions; although, as Einstein said about time², they are “very persistent” illusions. TDVP is built upon, and an extension of, the monumental works of a number of intellectual giants like Pythagoras, Fermat, Leibniz, Poincare, Cantor, and Minkowski; but most especially, it is built upon on the deep insights of Max Planck and Albert Einstein.

Max Planck said: "As a man who has devoted his whole life to the most clear-headed science, to the study of matter, I can tell you as the result of my research about atoms this much: There is no matter as such! All matter originates and exists only by virtue of a force. We must assume behind this force the existence of a conscious and intelligent Mind. This Mind is the matrix of all matter."¹⁴

And, Albert Einstein said: “Space-time is not necessarily something to which one can ascribe a separate existence.”¹⁵ And “I want to know the thoughts of God, everything else is just details”¹⁶

These statements, from two of the most brilliant scientists who spent their entire lives studying physical reality, reveal the important conclusion that the common perceptions of matter, energy, space, and time, conveyed to our brains by the physical senses, are subtle illusions! And both of them conclude that the reality behind these subtle illusions is a conscious, intelligent Mind!

It has long been known that the appearance of solid matter is an illusion, in the sense that there appears to be far more empty space than substance in an atom. But now we learn that the matter of sub-atomic particles and the “empty” space around them are also illusory. This is, however, consistent with quantum physics experiments that bear out the conclusion resulting from the resolution of the EPR paradox¹⁶ with the empirical demonstration of John Bell’s inequality¹⁷ by experimental physicist Alain Aspect¹⁸ and many others that the particles and/or waves of the objective physical reality perceived through our senses cannot be said to exist as localized objects until they impact irreversibly on a series of receptors constituting a distinct observation or measurement by a conscious entity.

We must be clear, however, that this does not validate subjective solipsist theories like that of Bishop Berkley¹⁹ as one might think; rather, it reveals a deeper, multi-dimensional reality, only partially revealed by the physical senses. It suggests that reality is like a fathomless, dynamic ocean that we can’t see except for the white caps. The difference is that the particles and waves, analogous to the white caps, only appear in response to our conscious interaction with the ocean of the deeper reality.

As noted above, Albert Einstein is quoted as saying: “Ich will Gottes Gedanken zu wissen, alles anderes ist nur Einselheit.” (I want to know God’s thoughts, the rest is just detail.) And he also said “Rafinert ist der Herr Gott, aber Bohaft ist er nicht!” (The Lord God is clever, but he is not malicious.) Taken together, these two statements reveal that Einstein’s science was rooted in a deeply spiritual understanding of reality. It appears that he believed that the universe, as a manifestation of God’s thoughts, is very complex, but understandable. Agreeing with Einstein, TDVP seeks to reveal that all things are, in fact connected to, and part of that deeper ocean of reality, only momentarily appearing to be separated from it. This apparent separation, perpetuated by the conscious drawing the distinction of ‘self’ from ‘other’ and the drawing of distinctions in self and other, allows us to interact with and draw distinctions in the ‘other’. TDVP posits that, although ostensibly separate in the 3S-1t world of our physical perceptions, we are never truly separated from the whole of reality, but remain connected at deeply embedded multi-dimensional levels.

There are some in the current mainstream of science who do see the universe as deeply mathematical, but even those scientists seem to shy away from including consciousness in their equations. An example is the Swedish physicist Max Tegmark. In his brilliant book “Our Mathematical Universe” he concludes that the ultimate nature of reality is mathematical structure. In reaching this conclusion, however, he strips mathematical description of any intent or purpose. He says “A mathematical structure is an abstract set of entities with relations between them. The entities have no ‘baggage’: they have no properties whatsoever except these relations.” ^{ref p. 231} In other words, he still does what most mainstream materialistic scientists do: he throws the baby out with the bath water. It is critically important to separate science from fantasy and wishful thinking, but consciousness is an extremely important part of reality and should not be excluded from the equations of science just because it complicates the picture.

From the broader viewpoint of TDVP, it is not surprising that mainstream science, focused, as it is, on the limiting philosophy of reductionist materialism, has lost touch its metaphysical roots, and thus cannot explain how it is that a large part of reality is not available to us for direct observation, but makes its existence known only indirectly through quantum phenomena like non-locality and quantum entanglement, as well as the near light-speed vortical spin of fermions and the effects of so-called dark matter and dark energy in the rotation of spiral galaxies²⁰.

TDVP also answers the real need to explain why we sometimes catch glimpses of a broader reality in rare extra-corporeal (out-of-body) experiences and other documented psi phenomena. The current mainstream scientific paradigm cannot explain so-called anomalous phenomena and the “missing” portions of reality because there is no place in its formulation for phenomena that may involve more than matter and energy interacting in three-dimensions of space and one dimension of time. TDVP, on the other hand, reveals a multi-dimensional reality and the need to recognize a third form of reality, not measurable as mass or energy, in the equations of science. As we shall see, TDVP provides a theoretical basis for a much deeper understanding of reality, as well as providing the appropriate tools for exploring it.

DO WE LIVE IN AN ACCIDENTAL UNIVERSE OF RANDOM COINCIDENCES?

Dividing the world of our experiences into the internal or subjective and the external, assumed to be completely independent of any form of consciousness, i.e. leaving consciousness out of the equations, as the current scientific paradigm does, alienates consciousness from the ‘real’ world of the physical universe and leads to an endless chain of unresolvable paradoxes. The prevalence of this attitude among scientists is expressed very well by MIT physicist - become science writer Alan Lightman in his recent book “The Accidental Universe”. In talking about the apparent ‘fine-tuning’ of the physical universe (if any one of a number of parameters were only a tiny bit different, there would be no chance for life as we know it), he says “Intelligent Design is an answer to fine-tuning that does not appeal to most scientists.” 

When confronted with the observer-related non-locality of Bohr’s solution to the EPR paradox, most scientists prefer the multiverse theory, devised to preserve Cartesian duality and keep consciousness out of the picture of ‘scientific objectivity’. In the multiverse theory, there are many, many parallel universes. Just how many there are is unknown and unknowable, because your consciousness only exists in this one, and unfortunately you cannot experience any of the other universes. Thus, just like the spate of string theories, there is no hope of proving or disproving such a theory. Even though these scientists pride themselves in being ‘hard-nosed’ objective scientists (read: materialists), it doesn’t seem to bother them that string theory and the multiverse theory cannot be tested. At best, they can only be internally consistent; and thus they do not even qualify as scientific hypotheses. By retreating into safely unprovable theories, they continue to throw the baby out with the bath water. TDVP, on the other hand, by including consciousness as an objective reality, is producing testable results and explaining observations that the current materialistic paradigm cannot explain. Several of these are listed in the previous section. In a new paper, I take the time to explain exactly how we put consciousness into the equations as part of objective reality, and show how doing so explains many things inexplicable in the current materialistic paradigm.

Unifying Quantum Physics and Relativity

The full unification of quantum physics and relativity is brought about in TDVP by applying the tools of CoDD and Dimensional Extrapolation²¹ to the mathematical expressions of three well-established features of reality, recognized in the current scientific paradigm: 1.) quantization of mass and energy as two forms of the same essential substance of reality; 2.) introduction of time as a fourth dimension, and 3.) the limitation of the velocity of rotational acceleration to light speed, c. In this process, the need for a more basic unit of quantization is identified, and when it is defined, the reason there is something rather than nothing becomes clear.

Einstein recognized that mass and energy are interchangeable forms of the physical substance of the universe, and discovered that their mathematical equivalence is expressed by the equation E=mc². In TDVP, accepting the relativistic relationship of mass and energy at the quantum level, we proceed, based on Planck’s discovery, to describe quantized mass and energy as the content of quantized dimensional distinctions of extent. This allows us to apply the CoDD to quantum phenomena as quantum distinctions and describe reality at the quantum level as integer multiples of minimal equivalence units. This replaces the assumption of conventional mathematical physics that mass and energy can exist as dimensionless points analogous to mathematical singularities.

The assumption of dimensionless physical objects works for most calculations in practical applications because our units of measurement are so extremely large, compared to the actual size of elementary quanta, that the quanta appear to be existing as mathematical singularities, i.e. dimensionless points. (The electron mass, e.g., is about 1x10^-30 kg, with a radius of about 3x10^-15 meter.) Point masses and point charges, etc. are simply convenient fictions for macro-scale calculations. The calculus of Leibniz and Newton works beautifully for this convenient fiction because it incorporates the fiction mathematically by assuming that the numerical value of a function describing the volume of a physical feature of reality, like a photon or an electron, can become a specific discrete finite entity as the value of a real variable, like the measure of distance or time approaches zero asymptotically (i.e. infinitely closely). This is a mathematical description of a non-quantized reality. But we exist in a quantized reality.

Planck discovered that the reality we exist in is actually a quantized reality. This means that there is a “bottom” to physical reality; it is not infinitely divisible, and thus the calculus of Newton and Leibniz does not apply at the quantum level. This is one reason scientists applying Newtonian calculus to quantum mechanics declare that quantum reality is ‘weird’. The appropriate mathematical description of physical reality at the quantum level is provided by the calculus of distinctions with the relationships between the measureable minimum finite distinctions of elementary particles defined by integral solutions of the appropriate Diophantine equations. The mathematics of quanta is the mathematics of integers.

In TDVP we find that, for quantized phenomena, existing in a multi-dimensional domain consisting of space and time, embedded in one or more additional dimensional domains, the fiction of dimensionless objects, a convenient mathematical expedient when we did not know that physical phenomena are quantized, is no longer appropriate. We can proceed with a new form of mathematical analysis, the calculus of dimensional distinctions (CoDD), and treat all phenomena as finite, non-zero distinctions. Replacing the dimensionless points of conventional mathematical physics with distinctions of finite unitary volume, we can equate these unitary volumes of the elementary particles of the physical universe with integers. We can then relate the integers of quantum reality to the integers of number theory and explore the deep relationship between mathematics and reality.

In TDVP, we have also developed the procedure of Dimensional Extrapolation using dimensional invariants to move beyond three dimensions of space and one of time. Within the multi-dimensional domains defined in this way, mass and energy are measures of distinctions of content. If there are other dimensions beyond the three of space and one of time that are available to our physical senses, how are they different, and do they contain additional distinctions of content? If so, how is such content different from mass and energy? We know that mass and energy are two forms of the same thing. If there are other forms, what is the basic “stuff” that makes up the universe? Is it necessarily a combination of mass and energy, - or something else? For the sake of parsimony, let’s begin by assuming that the substance of reality, whatever it is, is multi-dimensional and uniform at the quantum level, and that mass and energy are the most easily measurable forms of it in the 3S-1t domain. This allows us to relate the unitary measure of inertial mass and its energy equivalent to a unitary volume, and provides a multi-dimensional framework to explore the possibility that the “stuff” of reality may exist in more than two forms.

Since 1989, I have been determined to find a better way to explain putting the Primary Reality of Consciousness into the equations of science. In 1996 I published the book “Transcendental Physics”, an effort to make the 1989 work more accessible. It reached a few more people, but still only a relative handful of scientists and others interested in the merging of science and spirituality.  One who shared my vision, and has been my research partner for the past six years, is the world-renowned neuroscientist, Dr. Vernon Neppe, MD, PhD. Together Dr. Neppe and I have developed a comprehensive framework, a paradigm for the science of the future. We call it the Triadic Rotational Dimensional Distinction Paradigm (TDVP). It was first published as “Reality Begins with Consciousness” in 2011, and has been reviewed by more than 200 scientists and philosophers worldwide. We have also published a number of technical papers; and recently, I’ve found a way to explain the revelations of the Calculus of Distinctions of 1989, 1996 and 2011, in a more accessible way. This paper does that, and in the process, I believe, it does much more.

The exists empirical evidence that provides the answer to two important questions: 1.) Why is there something rather than nothing? And, 2.), what is missing from the current scientific paradigm? The answer to both questions can be summed up in one word: Consciousness. Without consciousness there could be no physical universe; and yet, there is no place in the current paradigm for consciousness. The clues that consciousness is the answer to the first question are plain in both relativity and quantum physics, but most mainstream scientists, steeped in reductionist materialism, are blind to those clues, and their belief – it is not even a valid scientific hypothesis – that the universe could exist without some primary form of the consciousness manifest in sentient life, is stubbornly maintained and the clues are ignored.

Many of the key scientists of the past were deeply spiritual. But the materialistic belief system widely taught in our educational institutions today brings otherwise rational people to scoff at, and ridicule, any mention of any form of intelligence superior to their own. This egotistical position of mainstream scientists is justified in their minds by the successes of materialistic science. But those successes lie almost entirely in the realm of explaining superficial physical mechanisms. Deeper and ultimately much more important questions about the meaning and purpose of manifest physical reality, life and conscious awareness, are beyond their reach. Those questions, of paramount importance to humanity, are within reach of meaningful analysis when consciousness is included in the equations of science. The purpose of a new paper by Close and Neppe is to show how this is done.

The bottom line is that, in this world of human experience, we will never truly understand the Nature of Reality until our searches for scientific and spiritual knowledge are merged into one serious, combined effort. Once this happens on a global scale, humanity will experience an explosion of new knowledge and understanding far beyond anything experienced so far in the current era of recorded history. The argument between creationists and evolutionists will be seen for what it iss: a profound misunderstanding. In TDVP, we show how consciousness is describable in the equations of quantum physics and relativity, and a few of the explanatory revelations produced as a result. And, believe me, this is only the tip of the iceberg! 

Note: the references indicated in this article will be provided in the paper on TRUE units, currently in press.