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.

Extended Pythagoras Theorem Using Hexagons

This article provides the geometric and algebraic proof of the variant equation of the Pythagorean theorem x^2-xy+y2=z^2 . The hypothesis that will be proven is that just as squares govern the original version x^2+y^2=z^2 , hexagons are found to govern x^2-xy+y^2=z^2 . Both the special case x=y&nbsp; and general case of x&ne;y&nbsp; are examined.

Deformation limit and bimeromorphic embedding of Moishezon manifolds

Let [Formula: see text] be a holomorphic family of compact complex manifolds over an open disk in [Formula: see text]. If the fiber [Formula: see text] for each nonzero [Formula: see text] in an uncountable subset [Formula: see text] of [Formula: see text] is Moishezon and the reference fiber [Formula: see text] satisfies the local deformation invariance for Hodge number of type [Formula: see text] or admits a strongly Gauduchon metric introduced by D. Popovici, then [Formula: see text] is still Moishezon. We also obtain a bimeromorphic embedding [Formula: see text]. Our proof can be regarded as a new, algebraic proof of several results in this direction proposed and proved by Popovici in 2009, 2010 and 2013. However, our assumption with [Formula: see text] not necessarily being a limit point of [Formula: see text] and the bimeromorphic embedding are new. Our strategy of proof lies in constructing a global holomorphic line bundle over the total space of the holomorphic family and studying the bimeromorphic geometry of [Formula: see text]. S.-T. Yau’s solutions to certain degenerate Monge–Ampère equations are used.

On parafermion vertex algebras of 𝔰𝔩(2) and 𝔰𝔩(3) at level −3 2

We study parafermion vertex algebras [Formula: see text] and [Formula: see text]. Using the isomorphism between [Formula: see text] and the logarithmic vertex algebra [Formula: see text] from [D. Adamović, A realization of certain modules for the [Formula: see text] superconformal algebra and the affine Lie algebra [Formula: see text], Transform. Groups 21(2) (2016) 299–327], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra [Formula: see text] of central charge [Formula: see text], and that [Formula: see text] is a direct sum of irreducible [Formula: see text]-modules. As a byproduct, we prove certain conjectures about the vertex algebra [Formula: see text]. We also obtain a vertex-algebraic proof of the irreducibility of a family of [Formula: see text] modules at [Formula: see text].

A Generic Algebraic Proof of the Unified Power Conservative Equivalent Circuit Theorem

This paper presents a generic algebraic proof of a recently published theorem [4], on the power conservative equivalent circuit for linear DC networks formed by time-invariant resistors and independent voltage and current sources. As the cited publication states, the internal losses of any network have two components: one variable and dependent on the internal resistances of the actual circuit and the power transferred to the pair of accessible terminals; and the other constant and dependent only on the internal voltage and current sources and the resistances of the actual network. It is also noted that the traditional Thévenin and Norton equivalent circuits are particular cases of the proposed equivalent circuit.

A Generic Algebraic Proof of the Unified Power Conservative Equivalent Circuit Theorem

This paper presents a generic algebraic proof of a recently published theorem [4], on the power conservative equivalent circuit for linear DC networks formed by time-invariant resistors and independent voltage and current sources. As the cited publication states, the internal losses of any network have two components: one variable and dependent on the internal resistances of the actual circuit and the power transferred to the pair of accessible terminals; and the other constant and dependent only on the internal voltage and current sources and the resistances of the actual network. It is also noted that the traditional Thévenin and Norton equivalent circuits are particular cases of the proposed equivalent circuit.