Vol. 4, No. 7, 2016, pp 504-506. doi: 10.12691/education-4-7-1 | Original Article

Abstract | |

1. | Introduction |

2. | The Guła’s Theorem |

3. | The Proof of Fermat’s Last Theorem For n=4 |

4. | Disproof the Birch and Swinnerton-Dyer Conjecture |

5. | The Proof of Goldbach’s Conjecture |

References |

The Guła’s Theorem. Only one in the world the proper proof of Fermat’s Last Theorem for n=4. Disproof the Birch and Swinnerton-Dyer Conjecture. The proof of Goldbach’s Conjecture.

**Keywords:** Birch and Swinnerton-Dyer Conjecture, Common Prime Factor, Diophantine Equations, Diophantine Inequalities, Fermat Equation, Greatest Common Divisor, Newton Binomial Formula

- Leszek W. Guła. Disproof the Birch and Swinnerton-Dyer Conjecture.
*American Journal of Educational Research*. Vol. 4, No. 7, 2016, pp 504-506. https://pubs.sciepub.com/education/4/7/1

- Guła, Leszek W.. "Disproof the Birch and Swinnerton-Dyer Conjecture."
*American Journal of Educational Research*4.7 (2016): 504-506.

- Guła, L. W. (2016). Disproof the Birch and Swinnerton-Dyer Conjecture.
*American Journal of Educational Research*,*4*(7), 504-506.

- Guła, Leszek W.. "Disproof the Birch and Swinnerton-Dyer Conjecture."
*American Journal of Educational Research*4, no. 7 (2016): 504-506.

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The Guła’s Theorem is dated 03-04 June 1997. The Fermat’s Last Theorem (FLT) is the famous theorem.

Disproof the Birch and Swinnerton-Dyer Conjecture (2009 Y.) we have on the strength of the Guła’s Theorem.

The Goldbach's Conjecture is one of the oldest and best-known unsolved problems in number theory and all

of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. ^{[4]}

**Theorem** **1** (Guła Theorem) *For each* *or** **for each* there exist finitely many pairs of positive integers such that:

where and and with even or with odd . ^{[2]}

**Theorem**** ****2**. For all such that and :

**Theorem** **3**.

^{[2]}

**Theorem 4** (FLT). For all and for all

**Remark 1.** Sufficient that we prove FLT for and for This is the remark 1.

**Remark** **2**. Fermat did not proved his own theorem for This is the remark 2.

**Remark** **3**. In we have the proof of FLT for . This is the remark 3.

**Remark** **4**. These hypothesis , **are different** because on the strength of theorem 1 for ;

For some relatively prime such that is positive and odd:

This is the remark 4.

**Proof.** Suppose that the equation has primitive solutions in .

Then it must be ^{[2]}

Without loss for this proof we can assume that

For some and for some such that and are co-prime:

Thus

Hence – For some such that are co-prime:

Therefore – For some such that and are co-prime:

Further it mus be – For some such that and are co-prime:

We assume that for some co-prime and for some :

which is inconsistent with This is the proof.

Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers to algebraic equations like

Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function near the point In particular this amazing conjecture asserts that if is equal to 0, then there are an infinite number of rational points (solutions), and conversely, is not equal to 0, then there is only a finite number of such points ^{[1]}.

**Conjecture 1** (Brich and Swinnerton-Dyer Conjecture) If is equal to 0:

then there are an infinite number of rational points

**Proof.** For all relatively prime and for some such that are co-prime:

**Disproof.** On the strength of the Theorems 1 and 2 –

If is not equal to 0, then – For all and for all and for all and for some such that , these two equations

have infinite numbers of such points in , namely or

If is not equal to 0, then – For all and for all relatively prime and for some such that , these two equations

have infinite numbers of such points in , namely or

If is not equal to 0, then – For all and for all relatively prime and for some such that , these two equations

have infinite numbers of such points in , namely or .

This is the disproof.

**Conjecture 2** (Goldbach Conjecture). For all and for some

**Proof**. The key of this proof are two common prime factors: 2 and 3

^{[2]}

This is the proof.

[1] | Claymath : – https://www.claymath.org/millennium/ | ||

In article | |||

[2] | Guła, L. W.: Several Treasures of the Queen of Mathematics – https://www.ijetae.com/files/Volume6Issue1/IJETAE_0116_09.pdf | ||

In article | |||

[3] | Narkiewicz, W.: WIADOMOŚCI MATEMATYCZNE XXX.1, Annuals PTM, Series II, Warszawa 1993. | ||

In article | |||

[4] | https://en.wikipedia.org/wiki/Goldbach%27s_conjecture. | ||

In article | |||