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SYNOPSIS

This is an improved version of an earlier treatise. It features a short-form proof that xⁿ + yⁿ = zⁿ is impossible in integers for n >2. It is the writers belief that this is Fermats original proof. This version addresses some of the comments from reviewers and resolves their objections.

To begin, a model for squared numbers will be introduced and used to devise a method to create all Pythagorean (x² + y² = z²) relationships. Equations will be derived from this process which indicate the existence of a Pythagorean equation in the model for squared numbers.

A model for higher powers of "n" will then be introduced. This model will be an extension of the model for squared numbers. Simple manipulations of this model will show that the "end game" packaging of quantities postulated to be xⁿ and yⁿ into spaces known to be xⁿ and yⁿ requires that x, y, and z form a Pythagorean equation ! This is totally incompatible with the postulation that xⁿ + yⁿ = zⁿ where n >2. The proof is thus Reductio ad Absurdum. . The recogniton of the afore-mentioned equations in the packaging process is the essence of the proof.

FORWARD

Validating Fermats assertion that he had a proof is of utmost importance. It will never be known for sure that he had a proof in his own day and age until some credible effort can be put together to substantiate this. This treatise will provide that proof.. Also, a short-form proof may provide valuable adjuncts to the recent accomplishments of "Twentieth Century Mathematics".

We have no direct knowledge of what might have been Fermats general approach; however, we have a few tantalizing clues from the notes he is said to have made in the margin of a text.

1. He was studying Diophantine analysis.

2. He conjectured about the possibility that the sum of two cubes could equal a third cube (in integers).

3. He rapidly asserted that this was impossible.

4. With equal rapidity he dismissed the possibility of an integral solution for xⁿ + yⁿ = zⁿ for any n > 2 !

5. He claimed to have a marvelous demonstration of this that was "Too long to include in the margin".
 
 

All of this bespeaks for a proof with the following characteristics:

1. It must be quite brief, perhaps only a few pages.

2. It was inspired by the Diophantine literature he was studying.

3. It required the recognition of a commonality in all xⁿ + yⁿ = zⁿ equations where n > 2. How else could he have proceeded so rapidly to his final statement?!!

4. It probably has several plateaus of logic which are simple in themselves, but obtuse in their application.

This treatise contains four sections which reconstruct what might have been Fermat's methodology in proceeding to a statement of his last theorem. These are:

1. Establishment of a model for squared integer numbers.

2. The use of this model to generate all Pythagorean (n =2) relationships and to identify equations which indicate the existence of a Pythagorean in x, y, and z when the equations occur in the model for squared numbers.

3. The creation of an advanced model in integers for xⁿ + yⁿ = zⁿ where n > 2.

4. The use of this model to show that packaging postulated values of higher powers of x and y into spaces known to be actual values results in a Pythagorean in x, y, and z; which is absurd and thereby proves the theorem.
 
 

GENERATION OF ALL PYTHAGOREAN TRIPLETS

This will explain the methodology for creating P. triplets from even "root numbers". To the knowledge of the writer, this method is entirely new. This has been confirmed by many math people. A simple model for squared numbers will be introduced. By manipulations of this model, it will be shown how a known Pythagorean triplet relates to an even "root number". Then an inverse process will be shown whereby any even "root number" can generate P. triplets. Depending on the factorability of the root number, a multiplicity of P. triplets can be generated from one root number.

This method generates all valid P. triplets. Trivial P. triplets such as 6, 8, and 10 having a common factor, ( in this case, 2), are avoided.
 
 

MODEL FOR SQUARED NUMBERS

Figure 1. Model for squared integer numbers

This model consists of a horizontal base of "ones" and an upper structure of "twos", thereby forming a triangular shape. The vertical and diagonal dimensions of the triangle equal the horizontal dimension. The cumulative value of the contents of the triangle equals the square of the horizontal dimension. Figure 1 shows the numbers from 1 to 8 and their squares. Observe that the cumulative value is actually the sum of odd numbers.Throughout this treatise, the model will be referred to as a "triangle". Sometimes it will be called an r, x, y, or z triangle depending on the horizontal dimension. Wherever meaningful, it may also be called an r², x², y², or z² triangle.

Figure 1(a).

The left figure shows how any diagonal segment of the model will represent a square. The overall triangle is 8², (64), and the shaded segment is 4², (16). The right figure shows how a vertical trapezoidal segment of the model can be represented by a slanted segment.
 
 

CHARACTERISTICS OF PYTHAGOREAN RELATIONSHIPS

Consider a known P. triplet, x = 8, y = 15, and z = 17 as shown in Fig. 2

Figure 2. From Pythagorean equation, 8² + 15² = 17².

The contents of the overall "z" triangle are z², (289). The dimension "y" below the figure is the dimension of the "y" triangle which is 15. The contents of this triangle are y², (225). Since x² = z²- y², the contents of the outlined trapezoid with width, (z-y), must equal x², (64).

Note the shaded rhomboid atop the trapezoid with height, (z-x), equal to 9, and width, (z-y), equal to 2. This rhomboid encloses a group of "twos" and has a total value of 2(9*2) = 36. For the trapezoid to form an x² triangle, the rhomboid contents must spill down as shown and form the small, shaded "r" triangle which abuts the lower portion of the trapezoid. The result is an x² triangle.

The small "r" triangle plus the lower portion of the trapezoid now equals x², (64). Note that "r", equals 6, and r² equals 36 which is the sum of the rhomboid contents, 2(9*2). "r" will always be even since r² is composed from the rhomboid contents which are all "twos". The following generalized equations are now obtained from the bottom of Figure 2.

(1) x = r + (z-y)
(2) y = r + (z-x)
(3) z = r + (z-x) + (z-y)
(4) r² = 2(z-x)(z-y) (since r² = rhomboid contents)
 
 

Figure 2(a). From Pythagorean equation, 8² + 15² = 17².

For additional clarity, the process is repeated in Figure 2(a) with the trapezoid equal to y². The trapezoid width is now (z-x). The rhomboid dimensions are still 9 by 2.

Again, the rhomboid contents spill down and form the small "r" triangle. It is the same size as before since the rhomboid dimensions are still (z-x) and (z-y). The numerical content of the "r" triangle is again 2(z-x)(z-y), 36. The "r" triangle abuts the lower portion of the trapezoid to form y². The generalized equations observed in Figure 2 are also observed from Fig. (2a).

The operations shown in Figures 2 and 2(a) have shown how the rhomboid contents in a known Pythagorean triplet "spill" down to form triangles whose contents equal r². P. triplets in x, y, and z can be created by a reverse process wherein an even number, r, is selected, squared to obtain (4), and factored to generate quantities representing the sides of rhomboids. Figure 3 shows this process in generalized form. The "ones" and "twos" have been omitted from Figure 3 for simplicity. The factors of r² are designated 2(z-x) and (z-y) to facilitate the explanation process using generalized terminology.
 
 

Figure 3. Method of generating P. triplets from even root numbers.

The shaded triangle, whose numeric content is r², now spills upward and forms a rhomboid of "twos" whose sides are (z-x) and (z-y) and whose numeric content equals that of the "r" triangle. Now a trapezoid equal to x² in numeric content has been created. A slanted trapezoid equal to y² with bottom dimension, (z-x), has also been generated. Note the interchangeability of the contents of the rhomboids and the r² triangle Note, also, that (1), (2), and (3) can be observed at the bottom of Figure 3. Figure 3 is the Framework for a Pythagorean triplet.

P. triplets in x, y, and z can be calculated by using equations (1), (2), and (3) with actual numbers in place of the generalized values, r, (z-x), and (z-y). Henceforth "r", will be called the "root number" since it is the starting point for creating P. triplets. Every even value of "r" will generate at least one P. triplet. If a given root number contains many prime factors, it will generate a multiplicity of P. triplets. The root number, 210, 2(3*5*7), generates 8 P. triplets.

The following process generates P. triplets in x, y, and z. Steps 1 thru 4 determine the rhomboid side dimensions, (z-y) and (z-x) as shown in Figure 3.

1. Select any even root number and express it in prime factors as shown.

Root number = r = 2(p₁p₂p₃p₄ . . . )

2. Square "r" to get the "root square" which is the contents of the "r" triangle and the rhomboid

Root square = r² = 4(p₁p₂p₃p₄ . . . )²

3. Obtain the the value of the rhomboid area (which is (z-y)(z-x) in Figures 2 and 3).

Since the rhomboids are composed of "twos", the root square must be halved to obtain the rhomboid area.

Rhomboid area = (Root square)/2 = 2(p₁p₂p₃p₄ . . . )²

4. Obtain the rhomboid sides by separating the rhomboid area into 2 factors. These will be (z-x) and (z-y) in general terminology. One side will be even since it contains 2 as a factor. The primes may be grouped in any combination. Only one is shown below.

Even factor = 2(p₁p₂ . . )²

Odd factor = (p₃p₄ . . )² .

5. Generate x, y, and z from (1), (2), and (3) by inserting actual numbers.

(1) x = 2(p₁p₂p₃p₄ . . . ) + (p₃p₄ . . )²

(2) y = 2(p₁p₂p₃p₄ . . . ) + 2(p₁p₂ . . )²

(3) z = 2(p₁p₂p₃p₄ . . . ) + (p₃p₄ . . )² + 2(p₁p₂ . . )²

In summary, equations (1), (2), and (3) are the result of r² = 2(z-x)(z-y). These equations were generated from the "Model for squared numbers" and should be used with this model throuighout the proof. Later this will be an important consideration in the proof of the theorem.
 
 

EXAMPLES OF PYTHAGOREAN TRIPLET GENERATION

The most primitive root number, 2(1*1), which is 2, yields only one solution:

root number	r squared	r squared/2	Even factor	Odd factor
2(1*1)	4(1*1)2	2(1*1)2	2(1)2	(1)2
x = 2(1*1) + (1)2	= 3
y = 2(1*1) + 2(1)2	= 4
z = 2(1*1) + 2(1)2 + (1)2	= 5

The root number, 2(2*1), which is 4, yields only one valid solution:

root number	r squared	r squared/2	Even factor	Odd factor
2(2*1)	4(2*1)2	2(2*1)2	2(2)2	(1)2
x = 2(2*1) + (1)2	= 5
y = 2(2*1) + 2(2)2	= 12
z = 2(2*1) + 2(2)2 + (1)2	= 13

The prime, 2, in the even factor cannot be transposed to the odd factor because x, y, and z would then have a common factor, 2. The result would be a 6, 8, 10 Pythagorean triplet which is a trivial doubling of the 3, 4, 5 solution.

The root number, 2(3*1), which is 6, yields two valid solutions since the prime, 3, may be transposed from the even to the odd factor.

Root number	r squared	r squared/2	Even factor	Odd factor
2(3*1)	4(3*1)2	2(3*1)2	2(3)2	(1)2
x = 2(3*1) + (1)2	= 7
y = 2(3*1) + 2(3)2	= 24
z = 2(3*1) + 2(3)2 + (1)2	= 25

Transposing the factor 3, the following valid solution results:

Root number	r squared	r squared/2	Even factor	Odd factor
2(3*1)	4(3*1)2	2(3*1)2	2(1)2	(3)2
x = 2(3*1) + (3)2	= 15
y = 2(3*1) + 2(1)2	= 8
z = 2(3*1) + 2(1)2 + (3)2	= 17

The above examples are the smallest P. triplets. As the number of primes in the root number goes up, the number of factors increases rapidly. Consider r = 2(1*3*5), which equals 30. This root number yields four P. triplets:

Even factors	Odd factors	x	y	z
2(1)2	(3*5)2	255	32	257
2(3)2	(5)2	55	48	73
2(5)2	(3)2	39	80	89
2(3*5)2	(1)2	31	480	481

The root number, 2(3*5*7), which equals 210, yields eight P. triplets:

Even factors	Odd factors	x	y	z
2(1)2	(3*5*7)2	11235	212	11237
2(3)2	(5*7)2	1435	228	1453
2(5)2	(3*7)2	651	260	701
2(7)2	(3*5)2	435	308	533
2(3*5)2	(7)2	259	660	709
2(3*7)2	(5)2	235	1092	1117
2(5*7)2	(3)2	219	2660	2669
2(3*5*7)2	(1)2	211	22260	22261

Note that the values of x, y, and z vary widely because the rhomboid changes shape with each different pair of factors. A long, slim rhomboid will result in large dimensions for z and either x or y.. As the rhomboid dimensions become more equal, the x and y values become more equal, and the z value becomes smaller. The rhomboid area is unchanged.

As the number of prime factors in "r" increases, the number of P. triplets doubles with each additional prime as follows:

"r" in prime factors	Numerical value of "r"	No. of P. triplets generated
2(1*3*5*7*11)	2310	16
2(1*3*5*7*11*13)	30030	32
2(1*3*5*7*11*13*17)	510510	64
2(1*3*5*7*11*13*17*19)	9699690	128

This method generates all valid Pythagorean triplets. Since any given P. triplet can be shown to have an even root number, "r", as shown in Figures 2 and 3, the P. triplet will eventually be generated as sequential even root numbers are selected and processed. The only limitations will be the capacity of the computational devices used.

This method is used in Dave ParkersÕ interactive Pythagorean triplet generator which appears in his website at: http://members.aol.com/parkerdr/math/. The P. triplet generator treats higher powers of individual prime numbers as prime numbers thereby avoiding trivial solutions.
 
 

MODEL FOR HIGHER POWERS

Consider the model for squares in Figure 1. A model for z³ is easily formed by stacking z² triangles horizontally as shown in Figure 4. Thus z³ is a wedge-shaped figure whose width, height, and length are all equal to z. For higher powers of z, the length is z^(n-2) while the width and height remain equal to z. For simplicity, the twos and ones are not shown in the model . It is understood that the bottom of the model consists of "ones" and the rest consists of "twos".
 
 

Figure 4. Model for higher powers of "z". (cubed value shown)
 
 

Figure 5. Relationships between x³, y³, and z³ in the advanced model

Figure 5 shows how y³ may represented as a "wedge" in the advanced model. If x³ + y³ = z³ is postulated to be true in integers, then x³ must be equal to the L-shaped form in Figure 5. Figure 6 shows this form separately.

Figure 6. x³ (postulated) form separated from the model.

Next, consider a separate z³ wedge with the wedge representing x³ removed, thereby leaving a void known to equal x³. Now suppose that the form in Fig. 6, postulated to be x³, is placed snugly on a level base within this x³void as shown in Figure 7

.

Figure 7. x³ L-shaped form placed within x³ void.

Note that parts of the L-shaped body (postulated to be x³ ) protrude above, to the side, and to the front of the void representing x³ ). Suppose that these protrusions are sliced away and the "detritus" is placed aside temporarily. The result is shown in figure 8.

Figure 8. x³ void partially filled in.

Figure 8 shows that the x³ void has been partially filled by x² triangles and a trapezoidal form of thickness, (z-y). The remaining space is a wedge-shaped void with dimensions "r". This void is thus composed of identical r² triangles. Note that the foregoing process has retained the "ones" on the bottom of the advanced model, thereby maintaining the integrity of the advanced model.Now, in keeping with the postulate that x³+ y³ = z³ in integers, the remaining "detritus" must exactly fit into the void of r² triangles to complete filling of the x³ void. The exact fit, of course, includes all of the "twos" and "ones" in their proper places. Note the dimensions, z, x, r, z-x, and z-y which have been identified as a result of the sequential filling of the x³ void.

Since the r² triangles are exactly alike, we may confine the discusssion to a single "z" triangle involving r, z, (z-x), and (z-y). When this is done the following equations emerge from Fig.8..

x = r + (z-y)
z = r + (z-x) + (z-y)

If the foregoing exercise is repeated with a postulated y³ L-shaped form filling a known y³ void, Figure 9 is obtained

Figure 9. y³ void partially filled in.

Again, in keeping with the postulate that x³+ y³ = z³, the remaining detritus must exactly fit into the void of r² triangles to complete filling of the y³ void. The following equations emerge:y = r + (z-x)
z = r + (z-x) + (z-y)

The value of "r" is the same as that found in Figure 8 since (z-x) and (z-y) have simply changed positions in the model. Again, we may confine the discussion to a single "z" triangle since the r, z, (z-x), and (z-y) dimensions are exactly the same in all of the triangles.

The two-stage filling of the x³ and y³ voids has yielded the following equations in integers which relate to a Pythagorean in x, y, and z.

(1) x = r + (z-y)
(2) y = r + (z-x)
(3) z = r + (z-x) + (z-y)
 
 

Figure 10. Side view of figure 9 showing rearrangement of sections..

View (a) of Figure 10 shows the side view of Figure 9 after the filling of the voids. View (b) shows how the shaded sections of view (a) can be rearranged [see Figure 1 (a)]. View (b) now has the form of Figure 3 which is the "framework" for the construction of a P. triplet.

Note that equations (1), (2), and (3) can be clearly seen in view (b). The "y" and "x" triangles overlap at the bottom of the figure. Note that the vertical dimension of "z" .may be traced up from the bottom of the figure, thru "r", z-y, and (z-x), thereby yielding z = r + (z-x) + (z-y). Note, also, the rhomboid of "twos" at the top of the figure which has the numerical content of 2(z-x)(z-y). The figure of view (b) will be revisited later in the proof.
 
 

Assuming that the "detritus" will fit exactly into the r³ void commits this proof to the "reductio ad absurdum" approach wherein the consequences of doing this will be examined. The result will be an enormous paradox which will prove the theorem.

The alternative to the above approach would be cutting and pasting the "detritus" to show that it cannot fit into the r³ void . This could be called the "direct" approach to a proof. This might conceivably work for the cases where "n" is equal to 3 or 4. However, beyond this it will be a hopeless slog into infinity.

It should be noted that the "Reductio ad Absurbum" approach shows what must be as the consequence of assuming that xⁿ + yⁿ = zⁿ exists in integers. Many reviewers have become diverted by examining "what if" situations before the proof has run its course. At this point, the following observations apply:
 
 

The integral nature of the model has been preserved.

Some reviewers have maintained that the proof has departed from an integral nature, but this is not true. The quantities x, y, z, r, and (z-x) and (z-y) are all integers . This has been meticulously preserved during the operations shown in Figures 4 thru 9. In all operations involving sections of the model for higher powers, the sections have been placed with the bottom side, consisting of "ones", on a level plane with the other sections. Figures 4 thru 9 show this clearly. The exact filling of the "r" voids has been done in like manner as the continuation of the postulate that xⁿ + yⁿ = zⁿ exists in integers. This is completely justified by the proof process wherein the consequences of this postulate will be examined..

The dimension, r, is an even number by the following logic:

Consider the postulate, xⁿ + yⁿ = zⁿ, where x, y, and z are integers with no common factor, 2.

Consider equation (1) , which is x = r + (z-y). This yields r = (x+y) - z.

1. If z is odd in xⁿ + yⁿ = zⁿ , then xⁿ + yⁿ (and x + y) must also be odd.

Therefore, r in r = (x+y) - z is: r = (odd ) - odd = even

2. If z is even in xⁿ + yⁿ = zⁿ, then xⁿ + yⁿ (and x + y) must also be even.

Therefore, r in r = (x+y) - z is: r = (even) - even = even

Therefore, r and each "r²" triangle will always be even for any postulated xⁿ + yⁿ = zⁿ in integers.

Each r² triangle is equal to a rhomboid of "2s". The content of the rhomboid will be a squared number.

This is true because when the "detritus" was postulated to completely fill the "r" voids in Figures 8 and 9 , the "detritus" had to be even and equal to all of the even r² triangles in the void. This is a most important consequence of the continuation of the postulate that xⁿ + yⁿ = zⁿ . Since "r" is an even integer, and the contents of each r² triangle is a square number, r² can be factored and used in the construction of a P. triplet as shown in Figure 3.

Equations (1), (2), and (3) are the result of generating a P. triplet from r² = 2(z-x)(z-y)

In this proof, all operations involving (1), (2), and (3) will be conducted within the framework of the model for squared numbers. This model occurs in figures 9 and 10 as a result of the sequential filling of the voids (in compliance with the postulate) and provides the following requisites for the construction of a P. triplet.

1. All of the quantities involved, x, y, z, and r are integers.

2. The quantity "r" is an even integer.

3. When (1), (2), and (3) are used with r² = 2(z-x)(z-y) to construct a P. triplet in the model for squared numbers, the result is an exact duplicate of the "z" triangles in Figures 8 and 9. The integers, x, y, and z form a P. triplet.

Equations (1), (2), and (3) are inexorably related to the model for squared numbers where they originated. P. triplets are generated in Figure 10 from r² = 2(z-x)(z-y). When equations (1), (2), and (3) are used outside of this model, that is, in the purely arithmetical sense, results occur which can mislead reviewers
 
 

As a consequence of the assumption that xⁿ + yⁿ = zⁿ in integers, the following has resulted:

1. The filling of the r³ voids has created individual "r" triangles which are all alike.

2. These "r" triangles are all even if xⁿ + yⁿ = zⁿ in integers (as shown earlier).

3. Each of the "z" triangles in Figures 8 and 9 is a model for squared numbers.

4. Equations (1), (2), and (3) in integers appear in each of the "z" triangles.

5. Using Figure 10 (which is identical to Figure 3) a P. triplet can be generated from (1), (2), and (3), and (4) [r² = 2(z-x)(z-y)] which is identical to the "z" triangles in figures 8 and 9.

Now in keeping with the geometry of Figure 3, a P. triplet in x, y, and z may be constructed in each "z". triangle. The numbers, x, y, and z which were postulated to form a cubed relationship in the advanced model must now form a x² + y² = z² relationship in the "z" triangles of the same model! This is clearly absurd.. If x² + y² = z², then z³ would greatly exceed x³ + y³ in the postulate. The cubed relationship is impossible in integers !!

The value of "n" in the postulation, xⁿ + yⁿ = zⁿ is completely immaterial !! This is because the proof process has introduced a general characteristic of all of the higher powers of n, namely, values of (z-x), (z-y), and r which, in the model, exist for any value of n !! The advanced models simply become longer as the value of "n" increases. The operations shown in Figures 4 thru 9 result in sections with similar shapes and characteristics for n > 3 as were observed for n = 3 !!!

Thus, the proof process will always yield the absurdity that x² + y² = z² no matter what value of n is postulated in xⁿ + yⁿ = zⁿ .

Therefore xⁿ + yⁿ = zⁿ is impossible in integers for n > 2 !!!

In a nutshell, this proof works because the proof process requires that x, y, and z in the relationship, xⁿ + yⁿ = zⁿ, must fit into a Pythagorean "framework". This framework is a "z" triangle within the model for advanced powers (n>2) of x, y, and z.