**The Beal Conjecture**? It is no longer a conjecture**, but theorem.**

Beal's conjecture is a conjecture in number theory. Billionaire banker Andrew Beal formulated this conjecture in 1993 while investigating generalizations of Fermat's last theorem.

The formulation of the hypothesis:

If ** ****where** ** ****- natural and** ** ****then** ** ****have a common prime factor.**

For the analytical solutions of the equation (Or ) Requires a new mathematical tool. This new mathematical tools developed based on the direct solution of Fermat's equation (See M. A. Eremin, "A new method for solving equations." Arzamas 2000).

Equation has a solution:

where - Criteria for the solvability of the equation in integers.

*u* - can have an infinite number of integer solutions.

where *t* - integer, *t> 0.*

(See M. A. Eremin, "What can a new method for solving equations Eremina (the problem solvability of Diophantine equations with different exponents)" Arzamas 2013)

From shows no at common divisor is 2 - is the minimum possible a prime number.

Data in can have an infinite number of positive integers.

A new mathematical tool makes it easy to solve equations of the form

We show the solution of these equations:

**Let us solve the equation** ** ****in integers.**

**Solution:**

Calculate the criteria for the solvability of the equation in integers;

at :

Find . Let . Then

Find .

Thus,

We write the solution of the equation

where - Whole number (> 0).

The exponents in numbers have an infinite set of values.

**The solution of the equation** ** ****in integers.**

**Solution:**

Calculate the criteria for the solvability of the equation ** in whole numbers: where .**

Let where - integer, .

Then

Thus,

We write the solution of the equation :

where

- integers greater than zero.

The exponents in numbers have an infinite set of values.

**The solution of the equation** ** ****in integers.**

**Solution:**

Calculate the criteria for the solvability of the equation ** in whole numbers for :**

Let Then ,

. let then . or .

Find .

where - integers.

The solution of the equation can be written as:

Where

- integers greater than zero.

**Find a solution** ** ****in integers.**

**Solution:**

Calculate the criteria for the solvability of the equation in integers:

at

Let Then ,

Let Then or: .

If then ,

. Let then: or .

If then ; , , ; then .

Find :

,

where ( ).

We write the solution of the equation :

,

where - integers.

- positive integers.

The exponents have an infinite set of values.

Number of formulas shows that these numbers are huge.

With the growth of the exponents in equation it becomes more difficult to calculate the criteria for the solvability of the equation in integers, as you have to produce more computational work.

**The following two theorems allow, without solving the equation** **Judge the solvability of the equation in integers.**

Theorem 1. Equation (Wherein - Natural numbers ) Has an infinite number of solutions in integers if GCD .

Examples: Greatest common divisor ,

Greatest common divisor .

Theorem 2. Equation (Wherein - Natural numbers), has an infinite number of solutions in integers,

if

or

Numbers have a common divisor.

Example 1: has a solution in integers, as and

where , - The minimum possible.

Example 2. Show under what equation , has a solution in integers.

Solution:

Thus, when equation has a solution in integers. Numbers for such equations are huge.

**M.A. Eremin.**