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.

Relation of Some Known Functions in terms of Generalized Meijer G -Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

METHODOLOGY OF THE TEACHING OF FINITE MATHEMATICS AT THE PEDAGOGICAL UNIVERSITIES DUE TO THE NEW TRENDS IN THE EVOLUTION OF THE MATHEMATICAL SCIENCES

There were introduced new methods of the teaching and instruction of the following parts of the Pre-calculus: (1) Binomial Series; (2) Trigonometry; (3) Partial Fractions. The problems introduced in the article for the Pre-Calculus Course in Finite Mathematics was developed by the author. These unabridged problems are developed within the new trends in the evolutions of the novelty of the syllabi in Mathematics due to the development of the Mathematics Sciences / Theory and Applications. These new trends in the Theory and Application of Mathematics Sciences have been added new demands to the newly revised textbooks and corresponding syllabi for the Mathematics Courses taught at the Junior two years Colleges and Pedagogical Universities.These newly developed problems are reflection of the Development of Mathematical and Engineering Sciences to offer great amount of learning conclusion/sequel to those who pursue a bachelor’s degree at the universities of the pedagogical orientation. The problems presented in the article here are developed and restructured in terms of the newly developed techniques to solve the problem in Finite Mathematics and Engineering sciences. Moreover, the techniques offered in the article here are more likely to get utilized in Advanced Engineering Sciences, too, within the content of the problems, which require to obtain finite numerical solutions to the Real Phenomena Natural Problems in Engineering Sciences and Applied Problems in Mathematical Physics.The aim of this present publication is to offer new advanced techniques and instructional strategies to discuss methodology and instructional strategies of the mathematical training of the students at the Pedagogical Universities. Moreover, these new teaching techniques and strategies introduced may be extended to the engineering sciences at the technical universities, too.The results and scientific novelty of the introduced methodology and learning conclusions and sequel of the new knowledge the students at the Pedagogical universities may be benefited from are in the following list of the learning conclusions, presented in the article here. The students of the pedagogical orientation may attain the mastery skills in the following sections of the combinatorics in Finite Mathematics subject:- The n! Combination of n different terms.- Evaluate the expressions with factorials.- Identify that there are -!!( )!nrnr various of combinations of r identical terms in n variations.- Identify and evaluate the combinatorial coefficients from the Binomial Theorem.- Identify and able to build the Pascal’s triangle of the binomial coefficients.- Utilize the Binomial Theorem to expand the binomial formula for any natural powers.- Utilize the Binomial Theorem to obtain the general formula for the n-th term of binomial expansion.- Utilize the Sigma Symbols in the Binomial Theorem for the n-th terms of the binomial expansion. 19- Generate the expansion for the power of the ex, where e is the base of natural logarithmic function y = f(x) = x.Practical significance: the methods of teaching and new teaching strategies offered here in the article alongside with the application of the new trends in the development of mathematical and mathematics education sciences can be useful for prospective and currently practicing teachers of mathematics. Moreover, the materials presented here in this article can be useful for the educational professionals in their professional development plans to improve the quality in education

Analysis of the Generalized Inverse Polynomial Scheme to Initial Value Problems

In this paper, we study the analysis of the generalized inverse polynomial scheme for the numerical solution of initial value problems of ordinary differential equation. At first, we generalize the scheme up to the fifth stage using the Binomial expansion and Taylor’s series method towards its derivation. The trend shows the generalization to the kth term. The analysis demonstrates the efficiency and the effectiveness of the generalized scheme.Keywords— Taylor’s Series, Initial Value Problem, Stability, Consistency, Convergence.

Series Representation of Power Function

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton&rsquo;s Binomial theorem and MacMillan Double Bi- nomial sum. Additionally, in section 4 exponential function&rsquo;s ex representation is shown.

Series Representation of Power Function

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton&rsquo;s Binomial theorem and MacMillan Double Bi- nomial sum. Additionally, in section 4 exponential function&rsquo;s ex representation is shown.