Fermat's Last Theorem: A Proof by Contradiction

In this paper I offer an algebraic proof by contradiction of Fermat’s Last Theorem. Using an alternative to the standard binomial expansion, (a+b) n = a n + b Pn i=1 a n−i (a + b) i−1 , a and b nonzero integers, n a positive integer, I show that a simple rewrite of the Fermat’s equation stating the theorem, A p + B p = (A + B − D) p , A, B, D and p positive integers, D < A < B, p ≥ 3 and prime, entails the contradiction, A(B − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 A i−1−j (A + B − D) j−1 # + B(A − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 B i−1−j (A + B − D) j−1 # = 0, the sum of two positive integers equal to zero. This contradiction shows that the rewrite has no non-trivial positive integer solutions and proves Fermat’s Last Theorem. AMS 2020 subject classification: 11A99, 11D41 Diophantine equations, Fermat’s equation ∗The corresponding author. E-mail: [email protected] 1 1 Introduction To prove Fermat’s Last Theorem, it suffices to show that the equation A p + B p = C p (1In this paper I offer an algebraic proof by contradiction of Fermat’s Last Theorem. Using an alternative to the standard binomial expansion, (a+b) n = a n + b Pn i=1 a n−i (a + b) i−1 , a and b nonzero integers, n a positive integer, I show that a simple rewrite of the Fermat’s equation stating the theorem, A p + B p = (A + B − D) p , A, B, D and p positive integers, D < A < B, p ≥ 3 and prime, entails the contradiction, A(B − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 A i−1−j (A + B − D) j−1 # + B(A − D) X p−1 i=2 (−D) p−1−i "X i−1 j=1 B i−1−j (A + B − D) j−1 # = 0, the sum of two positive integers equal to zero. This contradiction shows that the rewrite has no non-trivial positive integer solutions and proves Fermat’s Last Theorem.

On Two Problems Related to Divisibility Properties of z(n)

The order of appearance (in the Fibonacci sequence) function z:Z≥1→Z≥1 is an arithmetic function defined for a positive integer n as z(n)=min{k≥1:Fk≡0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn→∞(z(n+1)−z(n))/(logn)2−ϵ=∞, for all ϵ∈(0,2).

A plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem

We present a plane trigonometric proof for the case n = 4 of Fermat’s Last Theorem. We first show that every triplet of positive real numbers (a, b, c) satisfying a4 + b4 = c4 forms the sides of an acute triangle. The subsequent proof is founded upon the observation that the Pythagorean description of every such triangle expressed through the law of cosines must exactly equal the description of the triangle from the Fermat equation. On the basis of a geometric construction motivated by this observation, we derive a class of polynomials, the roots of which are the sides of these triangles. We show that the polynomials for a given triangle cannot all have rational roots. To the best of our knowledge, the approach offers new geometric and algebraic insight into the irrationality of the roots.

Extension of Fermat’s last theorem in Minkowski natural spaces

Minkowski natural (N + 1)-dimensional spaces constitute the framework where the extension of Fermat’s last theorem is discussed. Based on empirical experience obtained via computational results, some hints about the extension of Fermat’s theorem from (2 + 1)-dimensional Minkowski spaces to (N + 1)-dimensional ones. Previous experience permits to conjecture that the theorem can be extended in (3 + 1) spaces, new results allow to do the same in (4 + 1) spaces, with an anomaly present here but difficult to find in higher dimensions. In (N + 1) dimensions with $$N > 4$$ N > 4 there appears an increased difficulty to find Fermat vectors, there is discussed a possible source of such an obstacle, separately of the combinatorial explosion associated to the generation of natural vectors of high dimension.

DIAGNOSTICS OF MATHEMATICAL PROOF OF THE BEAL CONJECTURE IN MEDICAL PSYCHOLOGY (REMAKE OF PREVIOUS AUTHOR’S ARTICLES CONCERNING FERMAT’S LAST THEOREM)

In the given work diagnostics of mathematical proof of the Beal Conjecture (Generalized Fermat’s Last Theorem) obtained in the earlier author’s works was conducted and truthfulness of the suggested proof was established. Realizing the process of the Bill Conjecture solution, the mathematical structure defining hypothetical equality of the Fermat theorem was determined. Such a structure turned to be one of Pythagorean theorem with whole numbers. With help of Euclid’s geometrical theorem and Fermat’s method of infinite descent one can manage to set that Pythagorean equation in whole numbers representing Fermat’s Last Theorem cannot exist and then the Fermat theorem is true, that is Fermat’s equality in natural numbers does not exist. Thus mental scheme of “demonstratio mirabile”, which Pierre de Fermat mentioned on the margins of Diophantus’s “Arithmetic”, was reconstructed.

Fermat’s Last Theorem and Related Problems

Empirical evidence in support of generalizations of Fermat’s equation is presented. The empirical evidence consists mainly of results for the p = 3 case where Fermat’s Last Theorem is almost false. The empirical evidence also consists of results for general p values. The \pth power with respect to" concept (involving congruences) is introduced and used to derive these generalizations. The classical Furtw¨angler theorems are reformulated. Hasse used one of his reciprocity laws to give a more systematic proof of Furtw¨angler’s theorems. Hasse’s reciprocity law is modified to deal with a certain condition. Vandiver’s theorem is reformulated and generalized. The eigenvalues of 2p x 2p matrices for the p = 3 case are investigated. (There is a relationship between the modularity theorem and a re-interpretation of the quadratic reciprocity theorem as a system of eigenvalues on a finite-dimensional complex vector space.) A generalization involving generators and \reciprocity" has solutions for every p value.

Local criteria for the unit equation and the asymptotic Fermat’s Last Theorem

Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat’s Last Theorem to hold over F and also, for the nonexistence of solutions to the unit equation over F. For example, if two totally ramifies and three splits completely in F, then the asymptotic Fermat’s Last Theorem holds over F.

Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant [Formula: see text] such that the following holds. Let [Formula: see text] be a number field of degree [Formula: see text], let [Formula: see text] be any rational prime that is totally inert in [Formula: see text] and [Formula: see text] any elliptic curve defined over [Formula: see text] such that [Formula: see text] has potentially multiplicative reduction at the prime [Formula: see text] above [Formula: see text]. Then for every rational prime [Formula: see text], [Formula: see text] has an irreducible mod [Formula: see text] Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.