Proof of the Beal, Catalan and Fermat’s Last Theorem Based on Arithmetic Progression
Issue:
Volume 11, Issue 4, August 2022
Pages:
51-69
Received:
25 June 2022
Accepted:
18 July 2022
Published:
26 July 2022
DOI:
10.11648/j.pamj.20221104.12
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Abstract: The Beal’s Conjecture, Catalan’s Theorem and Fermat’s Last Theorem are cases within the number theory’s field, so the Beal’s Conjecture states that: AX+BY=CZ where A, B, C are positive integers and X, Y and Z are all positive integers greater 2, then A, B and C have a common prime factor and Catalan’s Theorem express that Yp=Xq+1 when X, Y, q, p are integer numbers and greater than one, this equation has not solution in integer numbers exception to 8, 9 numbers. The Fermat’s Last Theorem has the form of the Beal’s Conjecture when X, Y, and Z equal to n, then states that impossible find any solution in integer numbers for this equation. This article presents the proof of the Beal, Catalan and Fermat’s Last Theorem, and generalizes these theorems have relationship with arithmetic sequence that this sequence outcome from subtraction of exponent integer numbers between successive terms. Then illustrated an exponent integer numbers built from two parts: one of the progression and other the non-progression, when a Diophantine equation has square power we dealt with summation of one series of arithmetic sequence that can increase terms of a progression by other progression. Thus can find relationship between Pythagoras’ equation, Catalan and Fermat-Catalan’s equation that obtained from Pythagoras’ equation (a2+b2=c2), the other word Catalan and Fermat-Catalan’s equation a form of Pythagoras’ equation when displace a point on the circle, at that time Pythagoras’ equation reform to Catalan and Fermat-Catalan’s equation. And also the Beal’s Conjecture when A, B, C are coprime, another form of Fermat’s Last Theorem that both dealt with summation of several series of arithmetic progression, that impossible increase terms of a progression by other progression or a several series of sequence that shape is similar to a triangular that represented rows of progression and with a non-sequence parts that must change to sequences which rows of this progression less than initial progression. Also determine the Fermat’s Last Theorem has no solution in integer number then Beal’s Conjecture when A, B, C are coprime also has no solution in integer number. The last term provides rules of Beal’s Conjecture for solution and determine that this conjecture is super circles that obtained from primary circles, this primary circles existed from Catalan’s Theorem, Fermat-Catalan’s theorem and other forms. All of primary circles are based on the Pythagoras’ equation and right triangle.
Abstract: The Beal’s Conjecture, Catalan’s Theorem and Fermat’s Last Theorem are cases within the number theory’s field, so the Beal’s Conjecture states that: AX+BY=CZ where A, B, C are positive integers and X, Y and Z are all positive integers greater 2, then A, B and C have a common prime factor and Catalan’s Theorem express that Yp=Xq+1 when X, Y, q, p ar...
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The Analytical Solution of Some Partial Differential Equations by the SBA Method
Yanick Alain Servais Wellot
Issue:
Volume 11, Issue 4, August 2022
Pages:
70-77
Received:
19 September 2022
Accepted:
4 October 2022
Published:
17 October 2022
DOI:
10.11648/j.pamj.20221104.13
Downloads:
Views:
Abstract: Many phenomena in nature, especially in the current context of climate change, are modeled by nonlinear partial differential equations. Numerical methods exist to solve these equations numerically. But, the search for exact solutions, when they exist, is always necessary, in order to better explain the modeled phenomenon. The interest of the search for the exact solution results in the advantage of avoiding to analyze again the margins of errors, which sometimes, require the minimization. Thus, several methods are implemented to search for possible exact solutions. Despite the existence of various methods, difficulties have always surfaced. The case considered is that of strongly nonlinear partial differential equations. Thus, in the literature approached a new method called, the SBA method. In this paper, is used an analytical method called SOME-BLAISE-ABBO method (SBA method), to solve nonlinear partial differential equations. It is a method that is exclusively presented for the solution of exclusively nonlinear partial differential equations. The fundamental objective of this work is to show the effectiveness of the method for nonlinear problems. To this end, a reaction-convection-diffusion problem, a biological population model and a system of coupled Burgers’ equations are chosen to demonstrate the effectiveness, accuracy and efficiency of the said method. The easy obtaining of the exact solutions of these three chosen nonlinear problems allowed us to affirm the effectiveness of the method.
Abstract: Many phenomena in nature, especially in the current context of climate change, are modeled by nonlinear partial differential equations. Numerical methods exist to solve these equations numerically. But, the search for exact solutions, when they exist, is always necessary, in order to better explain the modeled phenomenon. The interest of the search...
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