Novel Approach of Existence of Solutions to the Exponential Equation

In this manuscript the exponential equation (3m + 3) x + (7m +1) y = z where m  Z in three variables for the occurrence of solutions belonging to the set of all integers or the concerned equation has no solution for various alternatives of m is investigated. Keywords— Exponential Diophantine equation, Pell equation, integer solutions


I.
Introduction Q In Mathematics, a Diophantine equation is a polynomial equation conventionally in two or more unknowns, such that only the integer solutions are sought or studied. Diophantine Analysis deals with numerous techniques of solving Diophantine equations in multivariable and multi-degrees. Suppose that a, b, c are pairwise co-prime positive integers. Then we call the equation , as an Exponential Diophantine equation. Nobuhiro Teraiq [1] proved that if , ,q q gratify then this equation has only the solution , provided that ) or . JuanliSu and Xiaoxue Liq [2], proved that if and , then the equation has only the positive integer solution by utilizing some results on the subsistence of primitive divisors of Lucas numbers. In this context one may refer [3][4][5][6][7][8]. In this paper, the exponential equation q is discussed for the existence of solutions in integers or this equation has no solution for different choices of m.

II. PROCESS OF TESTING THE HYPOTHESIS
The exponential equation for searching out solutions exists or not in integer is taken as q (1) where Investigate the hypothesis of (1) for the ensuing three cases.
(i) (ii) (iii) The possibilities of the above three cases are exemplified below.
(i) q (ii) , and (iii) , , and The detailed explanation of analyzing and all the above nine cases are given below.

Case (i): Suppose
These two values of q and direct (1) to the second-degree equation in two variables as follows (2) The very least roots of (2) are monitored by The other possible roots of (2) are located through the equivalent Pellian equation   Table (I). Case (ii): Let These two preferences of and deviate (1) to the equation occupying and as follows (6) The extremely least roots of (6) are checked manually and it is indicated by The alternative feasible solutions of (6) are sited through the indistinguishable equation (7) The pair flattering (3)q is computed by . The common solutions to (7) is communicated through the following equations for the convenience that Exploiting the formulae as there in case (i), the array of solutions for (6) existing in the set Z of all integers is concluded by (8) Table(III)q Case (iv): These supposition simplifies (1) to the fourth degree equation with two variables as declared below (14) Since, the square of an integer minus one can never be a square, the above postulation is always not possible. Consequently, the equation (14) and hence equation (1) does not possess a solution.

Case (v):
These hypotheses make things easier to (1) as the succeeding equation involving two variables with degree four (15) According to explanation given in case (iv), the statement produced above does not hold. As a result, the equation (15) and hence equation (1) does not have a solution.

Case (vi):
Repercussion of these selections reduces (1) to the equation consisting two unknowns with degree six as q (16) which can be modified by q (17) where ( The most promising solution to (17)  Hence, there does not exit a solution to (1) in integers.

Remark:
As a replacement of all other fraction for (25) also provide no solutions in integers to (1) Novel Approach Of Existence Of Solutions To The Exponential Equation (3m 2 + 3) x + (7m 2 +1) y = z 2