On the Non-Linear Diophantine Equation 61^x + 67^y = z^2 and 67^x + 73^y = z^2

Kumar, Sandeep; Gupta, Sani; Kishan, H.

doi:10.22457/apam.v18n1a13

Cited by 14 publications

(14 citation statements)

References 7 publications

(7 reference statements)

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“…Sroysang [10] discussed the Diophantine equation . Kumar et al [11] considered the non-linear Diophantine equations and . They showed that these equations have no non-negative integer solution.…”

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confidence: 99%

On the Exponential Diophantine Equation (132m) + (6r + 1)n = z2

Aggarwal

Kumar²

2021

J. Sci. Res.

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Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers, has no solution in the whole number.

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Section: ) ( )mentioning

confidence: 99%

On the Exponential Diophantine Equation (132m) + (6r + 1)n = z2

Aggarwal

Kumar²

2021

J. Sci. Res.

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“…In 2017, Asthana, S., and Singh, M. M. [3] studies the Diophantine Equation 3 + 13 = 2 and proved that thishas exactly four non-negative integer solutions for x, y and z. The solutions are (1, 0, 2), (1, 1, 4), (3,2,14) and (5,1,16) respectively.In 2018, Kumar et al [10] studied the non-linear Diophantine equations 61 + 67 = 2 and 67 + 73 = 2 . They proved that these equations have no non-negative integer solution.…”

Section: ι Introductionmentioning

confidence: 99%

On the Diophantine Equation 5 n x + 4 m p + 1 y = z 2

Moonchaisook¹,

Moonchaisook²,

Moonchaisook³

2021

IJRSI

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“…Burshtein [3] discussed all the solutions to an open problem of Chotchaisthit on the Diophantine equation 2 x + p y = z 2 when y = 1 and p = 7, 13, 29, 37, 257. Kumar, Gupta and Kishan [6] solved the Diophantine equation 61 x +67 y =z 2 and 67 x +73 y =z 2 and proved that the equations have not any non-negative integer solution. In this study, we discuss the Diophantine equation ሼሺ‫ݍ‬ ଶ ሻ ሽ ௫ + ‫‬ ௬ = ‫ݖ‬ ଶ where q is any prime number and p is an odd prime number.…”

Section: Introductionmentioning

confidence: 99%