View of Novel Approach of Existence of Solutions to the Exponential Equation (3m2 + 3) x + (7m2 +1) y = z2

Research Article

Novel

Approach

of

Existence

of

Solutions

to

the

Exponential

Equation

(3m

_{+ 3)}

_{+ (7m}

₊₁₎

_{= z}

1

_R.Vanaja

_,

_{V. Pandichelvi}

_q

1_{AssistantProfessor}

Departmentof Mathematics,

AIMANCollegeofArts&ScienceforWomen, (AffiliatedtoBharathidasanUniversity) Tiruchirappalli, Tamil Nadu

vanajvicky09@gmail.com

2_{AssistantProfessor}

PG &Research q DepartmentofMathematics,

UrumuDhanalakshmiCollege,

(AffiliatedtoBharathidasanUniversity), Tiruchirappalli, Tamil Nadu

mvpmahesh2017@gmail.com

Article History: Received: 11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021

Abstract— Inthis manuscript the exponentialequation (3m2 _{+ 3)} x _{+ (7m}2 ₊₁₎ y _{= z}2 _{where m}  _{Z inthree}

variablesfortheoccurrenceofsolutionsbelongingtotheset ofall integers or theconcernedequation has no solutionforvariousalternatives of misinvestigated.

Keywords— ExponentialDiophantineequation,Pell equation,integersolutions

I. Introduction

Q In Mathematics, a Diophantine equation is a polynomial equation conventionally in two or more

unknowns,suchthatonlytheintegersolutionsaresoughtorstudied. DiophantineAnalysisdealswithnumerous

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 𝑎, 𝑏,q 𝑐q gratify

then this equation has only the solution

, provided that ) or . JuanliSu and Xiaoxue Liq [2],

proved thatif 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 [3-8]. In this paper, the exponential equation

q isdiscussedfor theexistenceofsolutionsinintegersorthisequation hasnosolutionfordifferentchoicesof m.

II. PROCESS OF TESTING THE HYPOTHESIS

Theexponentialequationforsearchingoutsolutionsexistsornotinintegeristakenas q (1) where

Investigatethehypothesisof(1) fortheensuingthreecases.

(i)

(ii)

(iii)

The possibilitiesofthe abovethreecasesareexemplifiedbelow.

(i) q

(ii) , and

(iii) , , and

Case (i): Suppose

Thesetwovaluesof q and direct(1)tothesecond-degreeequationintwovariablesasfollows

(2)

Theveryleastrootsof(2)aremonitoredby

Theotherpossiblerootsof(2)arelocatedthroughtheequivalentPellianequation

(3)

Thelowestpositivevaluesofthecouple fulfilling(3)isestimatedby

. The solutionto(3)isgeneralizedbytheequations

,

Thesequenceofsolutionsfor(3) belongingtothesetZofallintegersisspecifiedasearlierbytheformulae

q

whichmeansthat

(4) q (5) where ,

Hence the numerous solutions to (1) for the preferred choices of as given in (5) are scrutinized by the

subsequentequation

)

The deduction of different types of equations for a variety of values of q and their corresponding solutions by utilizing (4) and (5) are tabularised in Table (I).

Case (ii): Let

Thesetwopreferencesof and deviate (1) totheequationoccupying and asfollows

(6)

The extremelyleastrootsof(6)arecheckedmanuallyanditisindicatedby

Thealternativefeasiblesolutionsof(6)aresitedthroughtheindistinguishableequation

(7)

Thepair flattering(3)q iscomputedby

.

Thecommon solutionsto(7)iscommunicatedthroughthefollowingequationsfortheconveniencethat

Exploitingtheformulaeasthere incase(i),thearrayofsolutionsfor(6)existinginthesetZofallintegersis

concludedby

(8)

where

,

Hence,theenormoussolutions to(1)forthe favourable choicesof as furnishedin(9) areinspectedbythe succeedingequation

)

The implication of nature of equations for a selection of values of and their resultant solutions by operating (8)q and (9) are tabularised in Table (II)

Case (iii): Consider

Undertheseassumptions, theparallelequationof(1)isderivedby

(10) Theextremelyfirstsolutionsto(10)aresupervisedbythecharacter

. Allotherpossiblesolutionsof(10)areperceivedthroughtheequation

(11)

Followingtheanalogousprocedureas mentionedincase(i)q andcase(ii)q byemployingtheprimarysolution andalsothegeneralsolutionsto(11),q thecycleof solutionsfor(10)tobethemembers ofthesetZofallintegersisdemonstratedbythesucceedingequations

(12) (13) where

,

Hence, the plentifulsolutions to (1) for the elected choices of as given in (13) are displayed bythe next equation

)

The inference of altered equations for the range of values of and their equivalent solutions by consuming (12) and (13) are tabularised in Table(III)q

Case (iv):

Thesesuppositionsimplifies(1) tothefourthdegreeequationwithtwovariablesasdeclaredbelow

(14)

Since,thesquareofanintegerminusonecanneverbeasquare, theabovepostulationisalwaysnotpossible. Consequently,theequation(14)andhenceequation(1) doesnotpossessasolution.

Case (v):

Thesehypothesesmakethingseasierto(1)asthesucceedingequationinvolvingtwovariableswithdegreefour

(15)

Accordingtoexplanationgivenincase(iv), thestatementproducedabovedoesnothold. Asaresult,theequation(15)andhenceequation(1)doesnothaveasolution.

Case (vi):

Repercussionoftheseselectionsreduces(1) totheequationconsistingtwounknownswithdegreesixas

q (16)

whichcanbemodifiedby

q (17)

where (

Themostpromisingsolutionto(17)ispointedoutby . Matchuptothevalueof with(18) establishedthat

.

whichisabsurdforintegervaluesof .

Theconclusionoftheproblemisequation(16) andhenceequation(1) doesnothaveasolutionwhen .

Note:

If , the set of all complex numbers, then the one and only one solution in integer to (1) is

Case (vii):

Influence oftheseoptionsreduces(1) totheequationconsistingtwounknownswithdegreefouras (19)

whichcanbereshuffledby

q where

(20)

Resolving theequivalentstructureoftheensuingproportionoftheequation(20)q

(21)

makeavailablewiththesolutionsas

(22)

(23)

Theonlyprospectofintegerfor q asmentionedin(22)is q forthenexttwoalternativesof and

(i) and ,q (ii)q and ,q

Forallothervaluesofaandb,itisnotedthat Theabovetwoselectionsof and forallvaluesof

certify (23) that and q respectively. Hence, the specified equation (1) is reduced into the

equationwithtwounknownsas

whichhasonlytwosolutions and

Remark:

Instead of taking the fraction (21)q as any other possible ratios also endow with the same

solution and q totheequation

Case (viii):

Thisspeculationsimplifies(1) tothefourth-degreeequationwithtwovariablesasaffirmedbelow

(24)

Modify(24)q intofactorsasgivenbelow

where

q (25)

The systematic procedure as in case (vii) for the proportion offers the

subsequentvaluesof and

(26)

(27)

Noneofthevaluesof and in(26)and(27)generate q and inintegers. Hence,theredoesnotexitasolutionto(1)inintegers.

Remark:

III. TABLES Table (I) Reduced form of (1) 1 271 (220326) x _{+ (514088)} y _{= z} 2 _(0,1,717) 2 4319 (55961286) x _{+ (130576328)} y _{= z} 2 (0,1,11427) 3 68833 (1.421394567 ×10 10₎ x ₊ (3.316587322 × 10 10₎ y _{= z}2 (0,1,182115) 4 1097009 (3.610286238 × 10 12₎ x ₊ (8.424001223 × 10 12₎ y _{= z} 2 (0,1,290241 3) 5 1748331 1 (9.169984906 × 10 14₎ x ₊ (2.139663145 ×10 14₎ y _{= z} 2 (0,1,462564 93) Table (II) Reduced form of (1) 1 30 (1,0,52) 2 112 (1,0,194) 3 418 (1,0,724) 4 1560 (1,0,2702) 5 5822 (1,0,10084) Table (III) Reduced form of (1) 1 17316 (899531571) x ₊ (2098906993) y _{= z} 2 (1,1,54758) 2 657552 (1.297123898 × 10 12₎ x ₊ (3.026622429 × 10 12₎ y _{= z} 2 (1,1,2079362) 3 2496966 0 (1.870451762 × 10 15₎ x ₊ (4.364387444 × 10 15₎ y _{= z} 2 (1,1,78960998 ) 4 9481895 28 (2.697190143 × 10 18₎ x ₊ (6.293443667 × 10 18₎ y _{= z} 2 (1,1,29984385 62) 5 3600623 24 (3.889346315 × 10 37₎ x ₊ (9.075141401 × 10 37₎ y _{= z} 2 (1,1,1.138617 043 ) IV. CONCLUSION

q Inthistext, theExponentialDiophantineequation , inthree

variableshasnumerousintegersolutions forparticular choices q andthe equationhas nosolutionforsome

other alternatives of is scrutinized. The conclusion is one can search the solutions to similar type of

exponential Diophantine equationhaving higher powers of and q than two by using various concept of

theoryofnumbers.

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