On the Diophantine equation 
                
                  
                
                $${y^{p} = \frac{f(x)}{g(x)}}$$
                
                  
                    
                      
                        y
                        p
                      
                      =
                      
                        
                          f
                          (
                          x
                          )

Subburam, S.; Togbé, Alain

doi:10.1007/s10474-018-0900-1

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2021

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“…where c 2 can be bounded using the linear form of the logarithmic method in Laurent, Mignotte, and Nesterenko [12], and an immediate estimation is The result of Hajdu, Laishram, and Tengely in [5] is much stronger than the following corollary. They explicitly obtained all solutions for the values k ≤ 10 using the MAGMA computer program along with two well-known methods (See Subburam [6], Srikanth and Subburam [13], and Subburam and Togbe [14]), after proving that n ≤ 19,736 for 1 ≤ k ≤ 10. Here, we have Corollary 1.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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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.

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