scholarly journals On the non-linear diophantine equation \({\boldsymbol{379}}^{\boldsymbol{x}}\boldsymbol{+}{\boldsymbol{397}}^{\boldsymbol{y}}\boldsymbol{=}{\boldsymbol{z}}^{\boldsymbol{2}}\)

2020 ◽  
Vol 4 (1) ◽  
pp. 397-399
Author(s):  
Sudhanshu Aggarwal ◽  
◽  
Nidhi Sharma ◽  
Keyword(s):  
Diophantine Equation ◽  
Existence Of Solution ◽  
Integer Solution ◽  
Negative Integer ◽  
Linear Diophantine Equation ◽  
Non Linear

In this article, authors discussed the existence of solution of non-linear diophantine equation \({379}^x+{397}^y=z^2,\) where \(x,y,z\) are non-negative integers. Results show that the considered non-linear diophantine equation has no non-negative integer solution.

scholarly journals Solutions to Non-Linear Diophantine Equation (5κ-1)χ+(5κ)y=z2 with k is Positive Even Integer

2019 ◽  
Vol 8 (2S7) ◽  
pp. 239-242
Keyword(s):  
Diophantine Equation ◽  
Research Methodology ◽  
Negative Integer ◽  
Linear Diophantine Equation ◽  
Single Solution ◽  
Congruence Theory ◽  
Non Linear ◽  
Two Stages

This research seeks for a solution (if any) to non-linear Diophantine equation 2 (5 1) (5 ) k x k y    z . There are 3 possibilities of solution to the non-linear Diophantine equation, which are single solution, many solutions, or no solution. The research methodology is conducted in two stages has t, which are using simulation to seek for a solution (if any) to non-linear Diophantine equation 2 (5 1) (5 ) k x k y    z and using Catalan’s conjecture and characteristics of congruence theory. It is proven that the non-linear Diophantine equation has single solution 2 ( , , ) (1,0,5 ) k x y z  for non-negative integer x, y,z and positive even of equal to or higher than

scholarly journals On exponential Diophantine equation 17x +83y = z2 and 29x +71y = z2

2021 ◽  
Vol 2070 (1) ◽  
pp. 012015
Author(s):  
Komon Paisal ◽  
Pailin Chayapham
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Integer Solution ◽  
Negative Integer ◽  
Exponential Diophantine Equation

Abstract This Diophantine is an equation that many researchers are interested in and studied in many form such 3x +5y · 7z = u2, (x+1)k + (x+2)k + … + (2x)k = yn and kax + lby = cz. The extensively studied form is ax + by = cz. In this paper we show that the Diophantine equations 17x +83y = z2 and 29x +71y = z2 has a unique non – negative integer solution (x, y, z) = (1,1,10)

scholarly journals On the Diophantine Equation 8^x+n^y=z^2

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Negative Integer

Let n be an positive integer with n = 10(mod15). In this paper, we prove that (1,0,3) is unique non negative integer solution (x,y,z) of the Diophantine equation 8^x+n^y=z^2 where x y, and z are non-negativeintegers.

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

2018 ◽  
Vol 18 (1) ◽  
pp. 91-94
Author(s):  
Satish Kumar ◽  
◽  
Sani Gupta ◽  
Hari Kishan
Keyword(s):  
Diophantine Equation ◽  
Linear Diophantine Equation ◽  
Non Linear

scholarly journals On Infinite Number of Solutions for one type of Non-Linear Diophantine Equations

2019 ◽  
Vol 9 (1) ◽  
pp. 1665-1669
Keyword(s):  
Infinite Number ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Prime Numbers ◽  
Periodic Sequence ◽  
Number Of Solutions ◽  
Linear Diophantine Equation ◽  
Linear Diophantine Equations ◽  
Non Linear

In this article, we prove that the non-linear Diophantine equation 𝑦 = 2𝑥1𝑥2 …𝑥𝑘 + 1; 𝑘 ≥ 2, 𝑥𝑖 ∈ 𝑃 − {2}, 𝑥𝑖′𝑠 are distinct and P is the set of all prime numbers has an infinite number of solutions using the notion of a periodic sequence. Then we also obtained certain results concerning Euler Mullin sequence.

scholarly journals Aplikasi Ring Kuadratik dalam Menyelesaikan Persamaan Pell

2020 ◽  
Vol 2 (2) ◽  
pp. 141
Author(s):  
Utami Priono ◽  
Wahidah Sanusi ◽  
Muhammad Abdy
Keyword(s):  
Diophantine Equation ◽  
Pell Equation ◽  
Linear Diophantine Equation ◽  
Pell Equations ◽  
Non Linear ◽  
Ring Method

Artikel ini membahas tentang penerapan Metode ring  (kuadratik) dalam mencari solusi pada persamaan Pell. Persamaan Pell merupakan bagian dari persamaan Diophantine non linear yang penyelesaiannya berupa bilangan bulat dengan bentuk umum persamaannya yaitu . Dalam penelitian ini persamaan Pell yang akan ditentukan solusinya yaitu . Metode yang digunakan dalam penelitian ini yaitu metode ring kuadratik. Metode ring kuadratik yang digunakan dalam menyelesaikan persamaan Pell memperhatikan konsep norm dan unit pada bilangan . Berdasarkan hasil penelitian, persamaan Pell positif  memiliki paling tidak satu solusi dengan nilai  yang dipilih. Sedangkan persamaan Pell negatif  tidak selalu memiliki solusi, hanya pada nilai  tertentu.Kata Kunci: Persamaan Pell, Ring Kuadratik, Norm This article discusses the application of the ring  (quadratic) method in finding solutions of the Pell equation. The Pell equation is part of the non linear Diophantine equation whose the solution is integer with the general form of the equation is . In this research, the Pell equation which the solution will be determined is . The method used in this research is the quadratic ring method. The quadratic ring method that will be used in solving the Pell equation takes the concepts of norm and unit in  number. Based on this research, positive Pell equations is             has at least one solution with the value of  that chosen. While the negative Pell equation is  doesn’t always have a solution, just at certain values of .Keywords: Pell Equation, Quadratic Ring, Norm.

scholarly journals Solutions to Non-Linear Diophantine Equation Pχ+ (P+5)y =z2 with P is Mersenne Prime

2019 ◽  
Vol 8 (2S7) ◽  
pp. 237-238
Keyword(s):  
Diophantine Equation ◽  
Research Methodology ◽  
Linear Diophantine Equation ◽  
Single Solution ◽  
Non Linear ◽  
Two Stages ◽  
Mersenne Prime

This research seeks for a solution (if any) to non-linear Diophantine equation 2 ( 5) x y p p z    with p is Mersenne prime. There are 3 possibilities of solution to the non-linear Diophantine equation, which are single solution, many solutions, or no solution. The research methodology is conducted in two stages, which are using simulation to seek for solution (if any) to non-linear Diophantine equation 2 ( 5) x y p p z    with p is Mersenne prime and using Catalan’s conjecture and characteristics of congruency theory. it is proven that the non-linear Diophantine equation has no solution for p  3.

scholarly journals On the Non-Linear Diophantine Equation p^x+〖(p+6)〗^y=z^2

2018 ◽  
Vol 18 (1) ◽  
pp. 125-128
Author(s):  
Sani Gupta ◽  
Satish Kumar ◽  
Hari Kishan
Keyword(s):  
Diophantine Equation ◽  
Linear Diophantine Equation ◽  
Non Linear
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