The Philosophical Meanings of Fermat's Last Theorem

Revised 1996

When Fermat's Last Theorem was unproven, it was very interesting. Once proven, it is no longer an interesting problem. For many, FLT was never fundamental or useful in any way. However, Fermat's last theorem points out that nature numbers are permanently entangled with (or confined to) the irrationals. Does not only the concept of permanent confinement of numbers provide a wonderful short proof for FLT, its philosophical meanings open up a new gate for mathematics.

I: Introduction

In 1640s, Pierre de Fermat wrote a note in the margin of his copy of the Arithmetica, "...it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, which however the margin is not large enough to contain."
For 350 years, the best mathematical minds in the world tried to reconstruct or rediscover Fermat's proof but all failed. This marginal note is now known as Fermat's Last Theorem.

II: Brief History

Fermat's equation (nth power of x (x^n) + nth power of y (y^n) = nth power of z (z^n)) is the combination of two equations: the Diophantine equation which must be solved in whole number, and the generalized Pythagorean equation which substitutes the 2nd power with n which can be all positive numbers. For the sake of not writing Fermat's equation repeatedly, I am giving it a unique name -- Hyper-Pythagorean-Diophantine equation or HPD equation in short. Thus, Fermat's last theorem can be restated as follow:
Fermat's Last Theorem -- There is no positive integer solution for HPD equation when n>=3.
Thus, if Fermat's last theorem is false, then we can and must find positive integer solution(s) for HPD equation when n>=3.
When n=2, HPD equation becomes traditional Pythagorean equation, and it has infinitely many whole number solutions, for example: 3^2 + 4^2 = 5^2, or 5^2 + 12^2 = 13^2, etc..
Although it is quite easy to experience that HPD equation does not have positive solution when n>=3 (for example, 2 cube + 3 cube = 35, but 4 cube = 64), it is an entirely different matter to prove that Fermat's last theorem is true for all n. Nonetheless, the only cases that need be proved are when n=4 or n is odd prime, because any integer greater than two is divisible either by 4 or by an odd prime.
Fermat himself proved the case of n=4 with the method of infinite descent. He started with a more general equation, 4th power of x + 4th power of y = 2nd power of z. The solution of this equation would give a Pythagorean triangle with two sides that were themselves squares which can form another Pythagorean triangle with smaller nonzero sides. Continuing in this manner, the smallest possible nonzero number (x=1, y=1) also must be a solution, but it is not. Therefore, the initial assumption must be false, and HPD equation has no positive solution when n=4.
In the next 350 years, some great mathematicians succeeded in proving Fermat's last theorem in a few special cases. Table 1 lists the solvers of those special cases.

Table 1: Solver of special cases
Year Solver Exponent
1640s Pierre de Fermat n = 4
1750s Leonhard Euler 3
1820s Peter G Lejeune Dirichlet
5
1850s Ernst E. Kummer n < 100
1993 Computer n < 4 million
With the help of computer, Fermat's last theorem is proved to be valid when n < 4,000,000. The chance to find a solution for HPD equation when n > 4 million is very small. It has twenty-six million zeros behind the decimal point. Nonetheless, the computer proof is not a valid mathematical reasoning; so Fermat's last theorem is still not proved in terms of mathematics.
On the mathematics front, two paths of significant progress on trying to prove Fermat's last theorem have opened up during the last 350 years. One path combines number theory with the theory of elliptic curve; the other combines number theory with Topology.
In 1840s, Gabriel Lame introduced a new concept -- algebraic numbers which are more general numbers than just whole numbers. In general, most algebraic numbers can be expressed as a product of primes in a unique way. By 1850s, Ernst E. Kummer modified and improved Lame's idea and proved that Fermat's last theorem is true for all n up to 100.
The further progress came from the modern approach to Diophantine equations -- the theory of elliptic curve, which is a perfect square equal to a cubic polynomial (y^2 = ax^3 + bx^2 + cx + d). In 1955, Taniyama made a conjecture -- that every elliptic curve is modular: the equation of the elliptic curve are divisible by a prime.
In 1980s, Gerhard Frey came up an idea to use the x^n and y^n of the HPD equation as perimeter to construct elliptic curves. So, if all elliptic curves must be modular (divisible by a prime), then HPD equation (associated with x^n, y^n) have positive integer solutions, and the Fermat's last theorem must be invalid. Thus, Frey's strategy provides a major machinery to test the validity of Fermat's last theorem in terms of elliptic curve theory and Taniyama conjecture.
In 1986, Kenneth A. Ribet proved that if the Taniyama conjecture is true, then Frey's elliptic curve can never exist. The task of proving Fermat's last theorem becomes either to prove that Taniyama conjecture is true or that Frey's elliptic curves cannot exist.
In 1993, Andrew Wiles announced that he had proved that a portion of the Taniyama conjecture (relevant to the Fermat's last theorem) is true.

III: Revisit the Issues

Fermat came up his last theorem by generalizing Pythagorean equation to higher power (greater than 2) and restricting its solution to be positive integer in terms of Diophantine equation. So, Fermat's last theorem is really an issue of prime number. Thus, I will give a brief visit on the theory of prime.
Prime numbers have two personality -- A) chaotic and B) somewhat ordered and confined.
A) Its chaotic nature:
• First, there are infinite number of prime.
Dirichlet theorem -- Every arithmetic sequence, a, a+b, a+2b, a+3b,..., a+nb,...,
Where a and b are relatively prime, contains an infinite number of primes.
Dirichlet theorem is much more powerful than Euclid's theorem on the infinitude of primes in the sequence of natural number, 1, 2, 3, ..., n, ....
• Second, the distribution of primes is also very chaotic. The prime distribution function D(x) represents the number of primes not exceeding x. Thus, D(4) is 2 because 2, 3 are prime and are smaller than 4. As x increases, the additional primes become scarcer and the problem was, what is the proper analytical expression for D(x)? Legendre, who had proved that no rational expression would serve, at one time give up hope that any expression for D(x) could be found.
B) The orderliness of primes:
• First, primes are the building blocks of the entire integers. There are two very important theorems on this.
Goldbach's conjecture: a) Every even integer is the sum of two primes. For example: 10 = 3 + 7, 26 = & + 19, etc.,
b) Every odd integer is either a prime or a sum of three primes.
1. Waring's theorem -- Every positive integer can be expressed as the sum of at most r kth powers, the r depending upon k.
For example: Every integer is either a cube or the sum of at most nine cubes, and every integer is either a fourth power or the sum of at most 19 fourth powers.
2. Second, primes not only are building blocks of the entire integers but have self-reflective orderliness. There are at least two theorems reflecting this.
1. Wilson theorem -- If and only if p is prime, the quantity (p-1)! + 1 is divisible by p.
2. Fermat's minor theorem -- If p is any prime and if a is any integer, then pth power of a (a^p) minus a is divisible by p.
3. Third, although the distribution of primes is very chaotic, it is nonetheless confined and bounded. Carl Friedrich Gauss used tables of primes to make conjectures about D(x) and inferred that D(x) differs little from the integration of (dt/log t) from 2 to x. He also knew that the ratio between the above integration to (x/log x) equals to 1 when x tends to infinity. His conjecture became the well-known Prime Number Theorem which was finally proved by Jacques Hadamard in 1896.
In 1850s, Pafnuti L. Tchebycheff also proved that there is always at least one prime between n and 2n - 2 when n>=3. Thus, the distribution of prime is, indeed, confined and bound although it is also very chaotic.
This paradoxical nature of prime (being both ordered and chaotic) is the heart of Fermat's Last Theorem. any approach (such as the Frey-Ribet-Wiles approach by using Taniyama conjecture on elliptic curve) trying to prove Fermat's last theorem without first to understand the meaning of this paradoxical nature of prime is doomed for failure. In fact, the entire number theory as it stands today will never be able to prove Fermat's last theorem because it conceives that naturals and irrationals are disjointed set, that is, the intersection set of naturals and irrationals is empty set.
In my book The Divine Constitution, I pointed out a paradox axiom -- every paradox always points out and leads to a higher truth. Thus, this paradoxical nature of primes must lead to a higher truth. Indeed, it does. Fermat's last theorem, in fact, points out that natural numbers are permanently entangled with (or confined to) the irrationals. But, what does this mean? And, how?

IV: The Internal Structure of Geometric Point

In quantum gravity, there are three big problems. First, there is an important difference between the graviton and the photon, that is, the photon couples only to charged particles, but the graviton couples to all particles, including the graviton itself. This kind of self-entanglement can give rise to infinity. Second, the quantum vacuum has a nonzero energy. Since gravity is supposed to interact with every form of energy and should interact with this vacuum energy. Thus, a vacuum would have a weight. On the one hand, quantum vacuum does not have weight in the real world. On the other hand, this quantum vacuum energy causes many gravity theories to produce meaningless answers -- infinities. Third, all theories seemed to diverge to infinity when they regard the fundamental particles as mathematical points.
Without the ability to address the issues of self-entanglement and of quantum vacuum energy, many physicists rushed to tackle the point-particle issue. They stretched the mathematical point into a geometrical string -- the so-called Superstring theory.
Although the superstring theory insists that the mathematical (geometrical) point has an internal geometrical structure, it does not know what kind of internal structure a mathematical point has. In my Prequark Chromodynamics, I insist that the internal structure of a mathematical point is not a sphere nor anything else but a donut, which can be described with 7 colors (3 quark colors, 3 generation colors and one colorless color).
In fact, there are many clues about the internal structure of mathematical and geometrical point even without the modern physics such as superstring or my Prequark Chromodynamics. There are clues in Topology. When we make a map by using North pole as the center, the entire outer edge (either to be a rectangle of a circle) represents a SINGLE point (South pole), that is, the single point and the infinite number of points 9the outer edge of the map) is identical in Topology. In this case to Topology, don't you think that there are some internal structures in mathematical or geometric point?
In the physical world (in physics), there are some physical properties associated with each mathematical or geometrical point. The collection of these points is a physical field. When two points in the physical field collide (that is, r [the distance between the two points] becomes zero), most of the physical properties associated with these points diverge into infinity. In fact, in the physical world, zero (r=0) and infinity are indistinguishable in the case of point particle. The mathematical point (which contains two collided physical field points) has very rich internal structure which consists of both nothingness (0) and infinity.
In mathematics, there is a relationship between 0 and infinity on the one hand, that is, infinity equals one (1) over zero (0). On the other hand, the fact that 0 and infinity are indistinguishable can also be clearly understood even in mathematics. When n approaches to infinity, the sequence 1, 1/2, 1/3, ..., 1/n,... approaches to 0. There is no way to distinguish from infinity by this sequence. This sequence reaches infinity and zero at the same time. that is, infinity and zero resides in the same point!!
But, why do all mathematicians (from past to present) have no clue whatsoever on the issue of internal structure of mathematical point? There are three major fallacies in mathematics. First, that the single point can be stretched into infinite points (as a circle) in Topology is viewed as a mathematical trick (as only convenience for mathematicians), not as an essence of mathematics.
Second, by definition, mathematical and geometrical points do not possess any physical property. All the misfortunes in physics on the point particle issue are the misfortunes of physicists, and it has nothing to do with mathematics.
Third, the greatest fallacy of mathematics is the definitions of LIMIT and CONTINUITY in the modern mathematics. The definition of LIMIT artificially equates a POINT with a big interval defined by epsilon and delta.
A few hundred years ago, Descartes, Euler and many others believed mathematics to be the accurate description of real phenomena, and they regarded their work as the uncovering of the mathematical design of the universe. Today, almost all mathematicians believe that mathematics is no longer absolute but arise arbitrarily. Thus, to equate a point to a giant interval defined by epsilon is perfectly legitimate in the modern mathematics, but this notion inevitably leads mathematics to the path of failure. The modern definition of LIMIT and CONTINUITY in mathematics have not only prevented mathematics reaching Ultimate Reality but have caused many unsolvable problems, such as the Fermat's Last Theorem.
The concept of continuity has a very important physical meaning. A string AB is continuous between point A and B if a pull force is applied on A and B, and string AB remains to be a single string. Thus, continuity can be defined in terms of physics. In physics, a point C in string AB is continuous if C is in touch with its neighboring points, that is, point C has at least two or more touching points in terms of physical bonding force. But, this concept of touching point cannot be defined in modern mathematics.
In mathematics, function f is continuous on an open interval (a, b) if every point c between (a, b) is a limit point of a big interval defined by epsilon and if f(c) is also a limit point of a big interval defined by delta. With this definition, for point C to be continuous, it must overlap with its neighboring point with a giant interval (defined by two epsilons) which contains an infinite number of points. On the one hand, this definition of continuity in continuous function) must be permanently confined to and entangled with its neighboring points, that is, every mathematical point has a very rich internal structure. On the other hand, this definition prevents mathematics of ever knowing that rich internal structure. That is, the touching point can never be defined in mathematics under the current concept of continuity.
Fermat's last theorem is not an isolated mathematical curiosity. It not only lies at the heart of number theory but points out the final secret that every number has an internal structure -- the naturals are inseparable from the irrationals. But, what is this internal structure?
I must introduce a new kind of number -- the touching numbers. Let b>0 and b<1. The distance between b and 1 is r. When r-->0, b-->1. In traditional mathematics, b=1 when r=0. With the concept of touching number, when b touches 1, that is r=0, b does not equal to 1. The traditional law of identity (a-b=0, then a=b) can no longer be applied in this region (the internal structure of a single number). Every number n on number line has at least two touching numbers, touching from each side. In fact, there are infinite amount of number which touches the number n from each side, but they can all be represented with a colored number (or a colored set). Thus, for every n, there are at least three numbers, and they can be expressed as n, n) or (n. The distances, r1=[n, n)], r2=[n, (n] and r3=[n), (n] are all equal to 0. Traditionally, n, n) and (n are utterly indistinguishable. But, we can actually color code these three numbers; such as: n as red, n) as yellow, and (n as blue.

V: Colored Numbers

Colored numbers are the result of the internal structure of mathematical point. Only with colored numbers does the internal structure of numbers become distinguishable. Since I discussed the concepts of colored numbers in details in my previous three books, I will only give a brief summary here.

1. Every number has an infinitely long tail. But, there are three different types of tails. The tail of 1/3 is infinitely long but predictable. The tail of number Pi is infinitely long and ultimately unpredictable. The tail of square root 2 is more predictable than the tail of Pi but less than the tail of 1/3. I used three colors (c, p, +) to identify these three different types of tails. So, 1/3 can be written as .3c, square root 2 = 1.414p, and Pi = 3.14159+. C means countable, + means uncountable, p is pseudo-uncountable.
2. Every tail has a ghost partner, that is, an infinity. C-tail corresponds to countable infinity, + tail to uncountable infinity, p-tail to pseudo-uncountable infinity. I used three colors {S(1), S(2), S(3)} to identify these three different types of infinities.
3. Zero (the Nothingness) is colorless. So, the entire number system consists of 7 colors {0, c, p, +, S(1), S(2), S(3)}. The entire rational numbers (except 0) is c-colored. Square root of 3 is p-colored. Pi is +colored. In this new number system, infinities are also numbers, colored numbers to be exact. Only by including infinities as number, the number system can be complete.

In 1922, Louis J. Mordell used a different approach (different from the Frey-Ribet-Wiles path) to tackle the Fermat's last theorem. Instead of trying to prove or disprove that theorem, he asked a different question -- how many solutions a Diophantine equation might have. Some Diophantine equations have infinitely many solutions -- such as the Pythagorean equation. Some have none, such as HPD equation when n>=3 and smaller than 4 million. He tried to work out what distinguished these possibilities. He noticed that if we look at all solutions of such an equation in complex numbers, then those solutions form a topological surface which has a finite number of holes, like a donut. The equations with infinitely many whole number solutions always had no hole, or just one, when solved in complex numbers. The number of holes on the surface corresponding to HPD equation is (n-1)(n-2)/2, and for n>3, there are always at least two holes. Thus, Mordell made a conjecture -- that equations that give rise to surface with two or more holes have only finitely many integer solutions.
In 1983, Gerd Faltings proved Mordell's conjecture, Soon afterward D. R. Health-Brown modified Faltings' approach to prove that the proportion of integers n for which Fermat's last theorem is true approaches to 100 percent as n becomes very large. Since the values of n in this approach was not specified, the proof was considered still incomplete.
Nonetheless, this Mordell's approach is much superior to the Frey-Ribet-Wiles path. Mordell's approach connotes the validity of colored numbers.

VI: A new Number Theory

Now, from the concepts of colored numbers and its associated torus (Topology), a new number theory can be constructed. Again, since I have described it in details in my books, I will only give a brief summary here.
In addition to the concept of colored numbers, I must introduce two more attributes of number -- softness and selfness. All finite number are somewhat rigid; 3 is always larger than 3-1. When the size of number increase, its rigidness reduces, such as the difference between three trillion and three trillion minus 1 becomes very insignificant. When the size of a number reaches infinity, it becomes utterly soft. We can squeeze infinitely more numbers into infinity without increasing its size. This ability to be itself is a very important attribute of infinities and numbers -- the SELF.

• Definition 1: r} is the ghost partner of a finite number r. r} is the number generated by removing the decimal point of r to the right side with a forever flying rocket. For example: 1/3}=333333...3333....
• Theorem 1: r} is an infinity.
• Definition 2: S is a SELF if Sth power of S is S itself (S^S=S). Example: the number 1 is a self.
• Definition 2a: If a <> b, and a * S(i) = b * S(i), then S(i) is a base color.
• Theorem 2: There are four SELF -- S(0)=1, S(1)=Countable infinity, S(2)=Pseudo-uncountable, S(3)=Uncountable.
• Theorem 2a: For i=1, 2 or 3, S(i) is a base color.
• Definition 3: C(i) is a colored tail if C(i) equals to -S(i)th power of 10.
• Theorem 3: 0, C(1), C(2), and C(3) are four base colors.
• Theorem 3a: There are seven base colors {0 (colorless), c, p, +, S(i), S(2), S(3)} in the number system.
• Theorem 4: All numbers (finite or infinite) can be represented by seven colors. Example:
• Pi=Pi}* -S(3)th power of 10 = 3.14159+
• Pi}= Pi * S(3)th power of 10 = 314159...+
• 1/3 = 1/3} * -S(1)th power of 10
• Theorem 5: These seven colors form a torus.
• The number line (including infinities) is isomorphic to a torus.
With the improved Mordell's conjecture, theorem 6 proved that Fermat's last theorem is true when n is large, but Fermat's last theorem can be proved with colored number theory without using Mordell's conjecture after the introduction of complementary rule and partner rule.
• Definition 4: If A(i) = +/- B(i)th power of 10, A(i) and B(i) are corresponding partner.
• Theorem 5a: For i= 1, 2 or 3, C(i) is corresponding partner of S(i)
• Definition 5: The elements of color set A is not partner among one another, A is a subset of the base color set of number system (defined in theorem 3a), and c is an element of A, set B= A-c, then c is the complement color of B.
• Definition 6: All integers are colorless numbers, rational numbers (excluding integers) are c-colored, all algebraic irrational numbers are p-colored, the nonalgebraic irrational numbers (such as Pi) are + colored.
• Theorem 7: N is an integer, N has three complementary numbers Nc, Np, N+. Note: In the traditional number theory, these four numbers N, Nc, Np and N+ are indistinguishable. That is, the distance between them is zero. And, Nc, Np, N+ each contains infinite amount of numbers.
• Theorem 7a: N - Nc = 0 and N <> Nc
Proof: N has colorless color by definition. Nc is c-colored and is a complement colored number of N. So, N < > Nc
Nc = N + ((-)S(1)th) power of 10
Nc - N = ((-)S(1)th) power of 10 = 0, S(1) is countable infinity. Although ((-)S(1)th) power of 10 = 0, it changes the color of N from colorless to c-colored.
• Theorem 8: The quotient between two different colored numbers will not have the same color as the denominator's.
• Theorem 9: The n roots of nth root of a number r (a complex number) equally divide a unit circle.
• Theorem 10: The number Pi is + colored.

The Final Proof of Fermat's Last Theorem

1. 2 Pi is divisible by 2. So, there are infinite whole number solutions for the HPD equation when n=2.
2. In order to have whole number solution for HPD equation when n >=3, 2Pi must be divisible by n per theorem 9. That is, Pi must be divisible by 2 (when n=4) or by all odd primes.
3. Per definition 6, all primes are colorless colored numbers.
4. Per theorem 10, Pi is + colored.
5. Per theorem 8, Pi divided by any prime cannot be a colorless colored number (whole number).