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However, this refutation is so surprising that we will not disclose it here, because it opens the way not only for the simplest proof of the FLT, but also automatically allows to reduce it to a very simple proof of the Beal conjecture. The disclosure the method of refuting this proof would cause a real commotion in the scientific world, therefore we will include this mystery among our riddles (see Appendix V Pt. 41).
So, we have demonstrated here solving to Fermat's problems (only by descent method!):
1) The proof of the Basic theorem of arithmetic.
2) The proof of the Fermat's theorem on the unique solving the
equation p3 = q2 + 2.
3) A way to prove Fermat's Golden Theorem.
4) A Fermat's way to solve the Archimedes-Fermat equation
Ax2 + 1 = y2.
5) The proof method of impossibility a3+b3=c3 in integers, which
opens a way to simplest proofs of the FLT and Beal conjecture.
6) A Fermat's proof his grandiose discovery about primes in the
form 4n + 1 = a2 + b2 which we have presented in another style in
Appendix IV, story Year 1680.
Over the past 350 (!!!) years after the publication of these problems by Fermat, whole existing science could not even dream of such a result!
3.5. Parity Method
Before we embarking on the topic "Fermat's Last Theorem" we note that this problem was not solved by Fermat himself using the descent method, otherwise in his FLT formulation there would be no mention of a "truly amazing proof", which certainly related to other methods. Therefore, to the above examples of the application of the descent method we will add our presentation of two methods unknown to today's science. The most curious of these is the parity method.
3.5.1. Defining Parity as a Number
The Basic theorem of arithmetic implies a simple, but very effective idea of defining parity as a number, which is formulated as follows:
The parity of a given number is the quantity of divisions this number by two without a remainder until the result of the division becomes odd.
Let's introduce the parity symbol with angle brackets. Then the expression ‹x› = y will mean:
the parity of the number x is equal to y. For example, the expression "the parity of the number forty is equal to three" can be represented as: ‹40›= 3. From this definition of parity, it follows:
– parity of an odd number is zero.
– parity of zero is infinitely large.
– any natural number n can be represented as n = 2w (2N – 1)
where N is the base of a natural number, w is its parity.
3.5.2. Parity Law
Based on the above definition the parity, it can be stated that equal numbers have equal parity. In relation to any equation this provision refers to its sides and is absolutely necessary in order for it to have solutions in integers. From here follows the parity law for equations:
Any equation can have solutions in integers if and only if the parities of both its sides are equal.
The mathematical expression for the parity law is WL = WR where WL and WR are the parities of the left and right sides of the equation respectively. A distinctive feature of the parity law is that the equality of numbers ca
3.5.3. Parity Calculation Rules
Parity of a sum or difference two numbers a and b
If ‹a› < ‹b› then ‹a ± b› = ‹a›.
It follows in particular that the sum or difference of an even and an odd number always gives a number with parity zero. If ‹a› = ‹b› = x then either ‹a + b› = x + 1 wherein ‹a – b› > x + 1
or ‹a – b› = x + 1 wherein ‹a + b› > x + 1
These formulas are due to the fact that
‹(a + b) + (a – b)› = ‹2a› = ‹a› + 1
It follows that the sum or difference of two even or two odd numbers gives an even number.
Parity of a sum or difference two power number an and bn
If ‹a› < ‹b› then ‹an ± bn› = ‹an›. If ‹a› = ‹b› = x then
only for even n:
‹an – bn› = ‹a – b›+ ‹a + b›+ x(n – 2) + ‹n› – 1
‹an + bn› = xn + 1
only for odd n:
‹an ± bn› = ‹a ± b› + x(n – 1)
When natural numbers multiplying, their parities are added up
‹ab› = ‹a› + ‹b›
When natural numbers dividing, their parities are subtracted
‹a : b› = ‹a› – ‹b›
When raising number to the power, its parity is multiplied
‹ab› = ‹a› × b
When extracting the root in number, its parity is divided
‹ b√a› = ‹a› : b
3.6. Key Formula Method
To solve equations with many unknowns in integers, an approach is often used when one more equation is added to the original equation and the solution to the original is sought in a system of two equations. We call this second equation the key formula. Until now due to its simplicity, this method did not stand out from other methods, however we will show here how effective it is and clearly deserves special attention. First of all, we note an important feature of the method, which is that:
Key formula ca
If this feature of the method is not taken into account i.e. add to the original equation some other one, then in this case, instead of solving the original equation we will get only a result indicating the compatibility of these two equations. In particular, we can obtain not all solutions of the original equation, but only those that are limited by the second equation.
In the case when the second equation is derived from the initial one, the result will be exhaustive i.e. either all solutions or insolvability in integers of the original equation. For example, we take equation z3 = x2 + y2. To find all its solutions we proceed from the assumption that a prerequisite (key formula) should be z = a2 + b2 since the right-hand side of the original equation ca
The product of numbers being the sum of two squares in all
cases gives a number also consisting the sum of two squares.
The converse is also true: if it is given a composite number being the sum of two squares then it ca
(a2+b2)(c2+d2)=(ac+bd)2+(ad−bc)2=(ac−bd)2+(ad+bc)2
Then from (a2+b2)(a2+b2)=(aa+bb)2+(ab−ba)2=(a2−b2)2+(ab+ba)2 it follows that the square of a number consisting the sum of two squares, gives not two decompositions into the sum of two squares (as it should be in accordance with the identity), but only one, since (ab−ba)2= 0 what is not a natural number, otherwise any square number after adding to it zero could be formally considered the sum of two squares.