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
• II: Brief History
• III: Revisit the Issues
• IV: The Internal Structure of Geometric Point
• V: Colored Numbers
• VI: A New Number Theory
• VII: The Final Proof of Fermat's Last Theorem

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.

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.

• 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.

• 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.

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.

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.

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).

Thus, Fermat's last theorem must be true for all n >=3.