The fibonacci sequence f n is defined by f 0 0, f 1 f 2 1 and f n f n. Duman and tuglu 2015, kitayama and shiomi 2017, remis 2011, wang and wang 2017, ye and zhang 2017. Pdf solutions of some diophantine equations using generalized. In this section, we recall some div isibility rules related to generalized fibonacci and lucas sequences u n a n d v n. Fibonacci numbers, lucas numbers, fibonacci matrix, diophantine equation. Fibonacci and lucas perfect powers by yann bugeaud, maurice mignotte, and samir siksek abstract this is the. Siksek, classical and modular approaches to exponential diophantine equations i. In the first part of this paper, some problems related to prime numbers and the role of diophantine equations in design theory is discussed. Lucas and fibonacci numbers and some diophantine equations, proc. Matijasevic 8 9 made use of divisibility properties of the fibonacci sequence to prove that every recursively enumerable set is diophantine.
Diophantine representation of fibonacci numbers over natural. Lucas and fibonacci number ans d some diophantine equations by j. Then we determine some identities involving these generalized sequences to present all solutions of the equations and x 2. On some diophantine equations journal of inequalities and.
Pdf integer solutions of some diophantine equations via. On diophantine equations involving lucas sequences. Pdf on nov 8, 2017, refik keskin and others published solutions of some diophantine equations in terms of generalized fibonacci and lucas numbers find, read and cite all the research you need. Fibonacci numbers for negative subscripts are defined as f. In paper 3 we investigate a particular form of the crossed ladders problem, nding many parametrized solutions, some polynomial, and some involving fibonacci and lucas sequences. Fibonacci numbers, lucas numbers, fibonacci matrix, diophantine. Balancing numbers were first considered by behera and panda in when they considered the integer solutions of the diophantine equation for some positive. Moreover, we obtain the set of positive integer solutions of some specific pell equations in terms of the integer sequences mentioned in the text. Let ln be the nth term of the lucas sequence defined by. Robbins, fibonacci and lucas numbers of the forms w2 1. Solutions of some diophantine equations in terms of. Considering the remainder modulo an integer is a powerful, foundational tool in number theory. Here, the case of the lucas numbers is more complicated since such a factorization of l. By using the generalized fibonacci and lucas sequences, we obtain all integer solutions of some diophantine equations such as x2kxyy2.
Ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. The contribution of fibonacci and lucas numbers to a. Then, we give all positive integer solutions of some diophantine equations in terms of. In this paper, we are interested in fibonacci and lucas numbers which are sum of three perfect powers of some prescribed distinct bases. Heptagonal numbers in the pell sequence and diophantine equations 2x2 y2. The fibonacci version of the brocardramanujan diophantine. Fibonacci and lucas numbers with applications thomas koshy. Ty jour t1 solutions of some diophantine equations in terms of generalized fibonacci and lucas numbers au bahar demirturk bitim, refik keskin y1 2019 py 2019 n1 do t2 hacettepe journal of mathematics and statistics jf journal jo jor sp 451 ep 459 vl 48 is 2 sn 2651477x2651477x m3 ur y2 2017 er. On the diophantine equation z n k n involving the order. Diophantine equations with products of consecutive terms. In this paper, we derive some identities on pell, pelllucas, and balancing numbers and the relationships between them. Fibonacci and lucas numbers which have exactly mafiadoc. Larger families of solutions to some diophantine equations. In mathematics, the fibonacci numbers, commonly denoted f n, form a sequence, called the fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1.
Keskin, integer solutions of some diophantine equations via fibonacci and lucas numbers, jis 12 2009. Solutions of some quadratic diophantine equations core. Keskin 4 exhibits families of solutions involving fibonacci and lucas numbers to some diophantine equations. Diophantine representation of fibonacci numbers over. However, it is still known that there are no prime solutions when n pelllucas, and balancing numbers and the relationships between them. On diophantine equations related to order of appearance in. I think the identities we present here will be found fascinating. In 12, the crucial point of the proof for the case of the fibonacci sequence was the existence of a factorizationof f n.
Pdf solutions of some diophantine equations in terms of. Then, we give all positive integer solutions of some diophantine equations in terms of these new sequences. Keywords fibonacci numbers lucas numbers generalized fibonacci numbers generalized lucas numbers diophantine equations. Diophantine equations with products of consecutive terms in lucas sequences. For example, in 11, marques and togbe showed that if s. We also deduce some formulas on the sums, divisibility properties, perfect squares, pythagorean triples involving these numbers. We study the problem of finding all integer solutions of the diophantine equations and using these equations, we also explore all integer solutions of some other diophantine equations. On some diophantine equations journal of inequalities. The fibonacci version of the brocardramanujan diophantine equation diego marques.
Yuri matiyasevich was able to show that the fibonacci numbers can be defined by a diophantine equation, which led to his solving hilberts tenth problem. We consider the sequences u n and v n which are the generalizations of fibonacci and lucas sequences, respectively. Three identities concerning fibonacci and lucas numbers. Heptagonal numbers in the associated pell sequence and diophantine equations x2. There are many papers in the literature which address diophantine equations involving fibonacci numbers. Some theorems and lemmas from now on, we will assume that n 0 and k 3 is odd. Positive integer solutions of some diophantine equations in. In this study, we give some new identities concerning fibonacci and lucas numbers.
Srinivasa rao 154781, new maruthinagar, dilsukhnagar, hyderabad500 060, a. The elementary properties of these numbers are easily established. Pdf diophantine equations with fibonacci and pell numbers. Since these rules a re proved i n, we omit their proofs. The fibonacci and the lucas polynomials are also important in a wide variety of research subjects djordjevic 2001, erkus. This function can be implemented in software mathematica. Congruences for fibonacci numbers pdf lecture notes, 2009 m. Diophantine equations, continuity fractions, binary recurrent sequences, fibonacci numbers, pell numbers, lower bounds for linear forms in logarithms. In this work, we determine all integer solutions of the diophantine equation and then we deduce the general terms of all balancing numbers 1. Solve integer equations, determine remainders of powers, and much more with the power of modular arithmetic.
Integer solutions of some diophantine equations via fibonacci and lucas numbers bahar demirturk and refik keskin department of mathematics sakarya university. Pdf we study the problem of finding all integer solutions of the diophantine. Integer solutions of some diophantine equations via. The lucas numbers satisfy remarkable difference equations, in some cases best expressed using fibonacci numbers, of which representative examples are the following. Pdf in this study, we deal with some diophantine equations. The proof is ineffective so in order to settle completely the above conjecture new ideas rather than just a long computation are needed. Equations with solution in terms of fibonacci and lucas. Pdf we study the problem of finding all integer solutions of the diophantine equations x2. Siksek, classical and modular approaches to exponential diophantine equations i, fibonacci and lucas perfect powers, annals of mathematics.
Using these equations, we also explore all integer solutions of some other diophantine equations. Solutions of some of the above equations are related to the fibonacci numbers. In this study, we give three identities concerning fibonacci and lucas numbers. Solutions of some quadratic diophantine equations sciencedirect. The tenth problem of hilbert was the problem of existence of an algorithm for deciding solvability of diophantine equations. In this study, we investigate positive integer solutions of the. On pell, pelllucas, and balancing numbers journal of. Positive integer solutions of some diophantine equations. In this paper, by using some congruences concerning with fibonacci and lucas numbers, we completely solve the diophantine equations ln 2lm x2, fn. This means that every positive integer can be written as a sum of fibonacci numbers, where any one number is used once at most. Moreover, we obtain the set of positive integer solutions of some specific pell equations in terms of the integer. On a diophantine equation involving powers of fibonacci numbers gueth, krisztian, luca, florian, and szalay, laszlo, proceedings of the japan academy, series a, mathematical sciences, 2020 extremality properties of some diophantine series rivoal, tanguy, experimental mathematics, 2010.
Heptagonal numbers in the associated pell sequence and diophantine equations x25x. A long standing problem asking whether 0, 1, 8 and 144 are the only perfect powers in the fibonacci sequence was recently confirmed by bugeaud, mignotte and siksek 1. Integer solutions of some diophantine equations via fibonacci. For example, marques 5 investigated the solutions of the fibonacci version of the brocardramanujan diophantine equation and showed that the diophantine equation 1. Positive integer solutions of some secondorder diophantine. Before proceeding further, some results will be needed in order to prove the theorem.
Diophantine equations involving fibonacci numbers have been no less popular, as documented in, d25 and in the historical sections of 8 and 7. Lucas and fibonacci numbers and some diophantine equations volume 7 issue 1 j. Fibonacci and lucas numbers, factorials, diophantine equations. Ams mathematics of computation american mathematical society. Classical and modular approaches to exponential diophantine. Moreover, we point out a connection between certain classes of the solutions and the pell numbers series. Moreover, there have been several papers attacking diophantine equations that involve both factorials and fibonacci numbers.
By using the generalized fibonacci and lucas sequences, we obtain all. Fibonacci and lucas perfect powers yann bugeaud, maurice mignotte, samir siksek abstract. There are many diophantine equations that have all solutions given in terms of particular lucas sequences as fibonacci and lucas numbers e. Lucas and fibonacci numbers and some diophantine equations. The contribution of fibonacci and lucas numbers to a quasiresidual metis design is explained.
Florian luca diophantine sets of fibonacci numbers. The fibonacci version of the brocardramanujan diophantine equation diego marques abstract. The fibonacci numbers are also an example of a complete sequence. In this note, we prove that the fibonacci version of the brocardramanujan diophantine equation n. Equations with solution in terms of fibonacci and lucas sequences. On some diophantine equations, journal of inequalities and.
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