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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 = 2^w (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 W_L = W_R where W_L and W_R 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 cannot be judged by the equality of their parity, but if their parities are not equal, then this certainly means the inequality of numbers.

      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

      ‹a^b› = ‹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 cannot be other as derived from the original equation.

      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 cannot be obtained otherwise than the product of numbers which are the sum of two squares. This is based on the fact that:

      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 cannot have prime factors that are not the sum of two squares. This is easily to make sure from the identity

      (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.

      However, this is not the case since there are numbers that cannot be the sum of two squares.

      As Pierre Fermat has established, such are all numbers containing at least one prime factor of type 4n − 1. Now from

      a²−b²=c; ab+ba=2ab=d; (a²+b²)²=c²+d²

      the final solution follows:

      z³=(a²+b²)³=(a²+b²)(c²+d²)=x²+y²

      where a, b are any natural numbers and all the rest are calculated as c=a²−b²; d=2ab; x=ac−bd; y=ad+bc (or x=ac+bd; y=ad−bc). Thus, we have established that the original equation z³=x²+y² has an infinite number of solutions in integers and for specific given numbers a, b – two solutions.

      It is also clear from this example why one of the Fermat's theorems asserts that:

      A prime number in the form 4n+1 and its square can be decomposed into two squares only in one way; its cube and biquadrate only in two; its quadrate-cube and cube-cube only in three etc. to infinity.

      4. The Fermat’s Last Theorem

      4.1. The Thorny Path to Truth

      4.1.1. The FLT up to now remains unproven

      The scientific world has been at first learned about the FLT after publication

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