Fermat's Last Theorem

Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation a n + b n= c for any integer value of n greater than two.

The theorem is called Pierre de Fermat's last because, of his many conjectures, it was the last and longest to be unverified. In 1630, Fermat wrote in the margin of an old Greek mathematics book that he could demonstrate that no integers (whole numbers) can make the equation x n + y n = z n true if n is greater than 2. A model for squared numbers will be introduced and used to devise a method to create all Pythagorean (x 2 + y 2 = z 2) 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 x n and y n into spaces known to be x n and y n requires that x, y, and z form a Pythagorean equation ! This is totally incompatible with the postulation that x n + y n = z n where n >2. 

After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents. In other words, it was necessary to prove only that the equation x n + y n = z n has no integer solutions (x, y, z) when n is an odd  prime number. This follows because a solution (x, y, z) for a given n is equivalent to a solution for all the factors of n. For illustration, let n be factored into d and e, n = de. The general equation x n + y n = z n  implies that (xd, yd, zd) is a solution for the exponent e
(x d) e + (y d) e= (z d) e.Thus, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for at least one prime factor of every n. All integers n > 2 contain a factor of 4, or an odd prime number, or both. Therefore, Fermat's Last Theorem can be proven for all n if it can be proven for n = 4 and for all odd primes p (the only even prime number is the number 2).

Pythagorean triples
A Pythagorean triple - named for the ancient Greek Pythagoras- is a set of three integers (a, b, c) that satisfy a special case of Fermat's equation (n = 2)  x 2 +y 2=z 2 Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,and methods for generating such triples have been studied in many cultures, beginning with the Babylonians and later ancient Greek, Chinese, and Indian mathematicians.The traditional interest in Pythagorean triples connects with the Pythagoreantheorem in its converse form, it states that a triangle with sides of lengths a, b, and c has a right angle between the a and b legs when the numbers are a Pythagorean triple. Right angles have various practical applications, such as surveying, carpentry, masonry, and construction. Fermat's Last Theorem is an extension of this problem to higher powers, stating that no solution exists when the exponent 2 is replaced by any larger integer.

Pythagorean equation
                                          
Diophantine equations
Fermat's equation, x n + y n = z n with positive integer solutions, is an example of a Diophantine equation named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equation. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:A=x+y  B=x 2+y 2

Fermat's Last Theorem: Proof for n=3

Theorem: Euler's Proof for FLT: n = 3

x 3 + y 3= z 3 has integer solutions -> xyz = 0

(1) Let's assume that we have solutions x,y,z to the above equation.

(2) We can assume that x,y,z are  coprime

(3) First, we observe that there must exist p,q such that:
(a) gcd(p,q)=1
(b) p,q have opposite parities (one is odd; one is even)
(c) p,q are positive.
(d) 2p*(p 2 + 3q 2) is a cube.

(4) Second, we know that gcd(2p,p 2+3q 2) is either 1 or 3. 

(5) If gcd(2p,p 2+3q 2)=1, then there must be a smaller solution to Fermat's Last Theorem n=3. 

(6) Likewise, if gcd(2p,p 2+3q 2)=3, then there must be a smaller solution to Fermat's Last Theorem n=3.
(7) But then there is necessarily a smaller solution and we could use the same argument on this smaller solution to show the existence of an even smaller solution. We have thus shown a condition of infinite descent.




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