scholarly journals On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

2020 ◽  
Author(s):  
Mahadi Ddamulira ◽  
Florian Luca
Keyword(s):  
Diophantine Equation ◽  
Exponential Diophantine Equation ◽  
Lucas Sequences

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.

The diophantine equation x2 + paqb = yq

2021 ◽  
pp. 1-18
Author(s):  
Hemar Godinho ◽  
Victor G. L. Neumann
Keyword(s):  
Diophantine Equation ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Lucas Sequences ◽  
Primitive Divisors

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

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
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