## вторник, 22 марта 2016 г.

### The proof (?) of Fermat's theorem

Mathematics considers natural numbers as points on a number line. I think it is a bit lopsided. Natural numbers are not just a series of points on a number line with an interval of 1. Let's try to build another series of intervals 3 ^ 0.5. All points of this series - the irrational numbers. But we still need the natural numbers to find the values of these points: 2 *3 ^ 0.5, 3 * 3 ^ 0.5, 4 * 3 ^ 0.5 ... .. Natural numbers - action. Natural numbers describe the algorithm of actions. Therefore, while solving the problem with natural numbers it is advisable to use the language of action and algorithms.

We’ll need an axiom. This axiom connects philosophy (epistemologywith mathematics.
Axiom: If there is an infinite number of examples of any relation on the infinite set, then all these examples can be combined (created)by one algorithm. This algorithm includes only pre-defined operations and relations for a given set, and corresponds to Gödel's incompleteness theorem.

Condition (1) determines an infinite number of examples of addition on the set if at least one example (of the decision) exists.

Theorem

f (x) - function corresponding to the following condition:
f (qx) = df (x) (1)
q, d - integers

A – the set which is given by the function f (x) (x - an integer). The addition operation is defined on this set (infinite number of simple solutions) only if there is n (n-integer nonzero number), for which:
f (n) + f (n + 1) = f (n + 2) (2)
f (n), f (n + 1), f (n + 2) - nonzero number

If the set has three consecutive numbers which satisfy the condition (2), the equation (3) has an infinite number of solutions:
f (a) + f (b) = f (c) (3)
a, b, c - integers.
f (a), f (b), f (c) - nonzero

If set A has three consecutive numbers which satisfy the condition (1),then set A is not the solution of equation (3).

Proof:

If there is at least one solution (3), then subject to the condition (1)there will be an infinite number of solutions. Consequently, according to Axiom, there will exist an algorithm.

The algorithm should be based on pre-defined operations. On set A only successor function and multiplication operation are pre- defined. Incidentally, multiplication and addition are not connected on this set in a similar way as on the set of natural numbers.

Therefore, the parental trio must exist. For example: 1 + 2 = 3 3 ^ 2 + 4 ^ 2 = 5 ^ 2, 3 ^ 3 + 4 ^ 3 + 5 ^ 3 = 6 ^ 3
Conversely, if condition (1) is true and there is no parental trio, there are no solutions at all.

Corollary1
If set A which satisfies the condition (1) is a solution of equation (3) in a nonzero integer numbers, it has an infinite number of solutions, and has the solution in the form of three consecutive numbers (parental trio).

Corollary 2
The function g (x) = x ^ r (r> 2) satisfies the condition (1), but set A does not satisfy the condition (2). Therefore, the equation a ^ r + b ^ r = c ^ r (r> 2) has no solution in positive integers.