scholarly journals Solving an exponential Diophantine equation

2010 ◽  
Vol 144 (4) ◽  
pp. 315-322
Author(s):  
Mario Huicochea
Keyword(s):  
Diophantine Equation ◽  
Exponential Diophantine Equation

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Solutions for a Class of the Exponential Diophantine Equation

2013 ◽  
Vol 753-755 ◽  
pp. 3149-3152
Author(s):  
Yin Xia Ran
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Algebraic Number Theory ◽  
Necessary Condition ◽  
Exponential Diophantine Equation ◽  
Elementary Method ◽  
Integer Solutions

We studied the Diophantine equation x2+4n=y11. By using the elementary method and algebraic number theory, we obtain the following conclusions: (i) Let x be an odd number, one necessary condition which the equation has integer solutions is that 210n-1/11 contains some square factors. (ii) Let x be an even number, when n=11k(k≥1), all integer solutions for the equation are(x,y)=(0,4k) ; whenn=11k+5(k≥0) , all integer solutions are(x,y)=(±211k+5,22k+1); when n≡1,2,3,4,6,7,8,9,10 the equation has no integer solution.

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