An exponential Diophantine equation related to the sum of powers of two consecutive k-generalized Fibonacci numbers

2014 ◽  
Vol 137 (2) ◽  
pp. 171-188 ◽  
Author(s):  
Carlos Alexis Gómez Ruiz ◽  
Florian Luca
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Exponential Diophantine Equation ◽  
Generalized Fibonacci Numbers

scholarly journals A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1010 ◽  
Author(s):  
Ana Paula Chaves ◽  
Pavel Trojovský
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Generalized Fibonacci Numbers ◽  
Quadratic Diophantine Equation ◽  
Positive Integers

The sequence of the k-generalized Fibonacci numbers ( F n ( k ) ) n is defined by the recurrence F n ( k ) = ∑ j = 1 k F n − j ( k ) beginning with the k terms 0 , … , 0 , 1 . In this paper, we shall solve the Diophantine equation F n ( k ) = ( F m ( l ) ) 2 + 1 , in positive integers m , n , k and l.

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.

SOME ALGEBRAICAL, COMBINATORIAL AND ANALYTICAL PROPERTIES OF PARAGRASSMANN VARIABLES

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4797-4819 ◽  
Author(s):  
MATTHIAS SCHORK
Keyword(s):  
Differential Equations ◽  
Fibonacci Numbers ◽  
Delta Function ◽  
Geometrical Interpretation ◽  
Generalized Fibonacci Numbers ◽  
Analytical Properties ◽  
Witten Index ◽  
Grassmann Variables

Some algebraical, combinatorial and analytical aspects of paragrassmann variables are discussed. In particular, the similarity of the combinatorics involved with those of generalized exclusion statistics (Gentile's intermediate statistics) is stressed. It is shown that the dimensions of the algebras of generalized grassmann variables are related to generalized Fibonacci numbers. On the analytical side, some of the simplest differential equations are discussed and a suitably generalized Berezin integral as well as an associated delta-function are considered. Some remarks concerning a geometrical interpretation of recent results about fractional superconformal transformations involving generalized grassmann variables are given. Finally, a quantity related to the Witten index is discussed.

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