Pdf integer solutions of some diophantine equations via. Equations with solution in terms of fibonacci and lucas. There are many papers in the literature which address diophantine equations involving fibonacci numbers. Integer solutions of some diophantine equations via. Equations with solution in terms of fibonacci and lucas sequences. This is the second in a series of papers where we combine the classical approach to exponential diophantine equations linear forms in logarithms, thue equations, etc. Diophantine equations with products of consecutive terms in lucas sequences. Keskin 4 exhibits families of solutions involving fibonacci and lucas numbers to some diophantine equations. 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.
Diophantine representation of fibonacci numbers over natural. Congruences for fibonacci numbers pdf lecture notes, 2009 m. For example, in 11, marques and togbe showed that if s. Pdf diophantine equations with fibonacci and pell numbers. The contribution of fibonacci and lucas numbers to a quasiresidual metis design is explained. The fibonacci and the lucas polynomials are also important in a wide variety of research subjects djordjevic 2001, erkus. Diophantine representation of fibonacci numbers over. Then, we give all positive integer solutions of some diophantine equations in terms of. Pdf solutions of some diophantine equations in terms of. Some theorems and lemmas from now on, we will assume that n 0 and k 3 is odd.
The fibonacci sequence f n is defined by f 0 0, f 1 f 2 1 and f n f n. On pell, pelllucas, and balancing numbers journal of. Fibonacci and lucas numbers which have exactly mafiadoc. 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. However, despite the ease with which many such properties are proved, there are a number. Robbins, fibonacci and lucas numbers of the forms w2 1. Heptagonal numbers in the associated pell sequence and diophantine equations x2. Then we determine some identities involving these generalized sequences to present all solutions of the equations and x 2. 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 also deduce some formulas on the sums, divisibility properties, perfect squares, pythagorean triples involving these numbers. Integer solutions of some diophantine equations via fibonacci. 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. 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. Solutions of some quadratic diophantine equations sciencedirect.
Srinivasa rao 154781, new maruthinagar, dilsukhnagar, hyderabad500 060, a. Positive integer solutions of some diophantine equations in. Considering the remainder modulo an integer is a powerful, foundational tool in number theory. Before proceeding further, some results will be needed in order to prove the theorem. The fibonacci numbers are also an example of a complete sequence. Matijasevic 8 9 made use of divisibility properties of the fibonacci sequence to prove that every recursively enumerable set is diophantine. In this paper, by using some congruences concerning with fibonacci and lucas numbers, we completely solve the diophantine equations ln 2lm x2, fn.
Keskin, integer solutions of some diophantine equations via fibonacci and lucas numbers, jis 12 2009. Moreover, there have been several papers attacking diophantine equations that involve both factorials and fibonacci numbers. Heptagonal numbers in the pell sequence and diophantine equations 2x2 y2. In this study, we investigate positive integer solutions of the. This means that every positive integer can be written as a sum of fibonacci numbers, where any one number is used once at most.
In this work, we determine all integer solutions of the diophantine equation and then we deduce the general terms of all balancing numbers 1. I think the identities we present here will be found fascinating. Moreover, we point out a connection between certain classes of the solutions and the pell numbers series. The proof is ineffective so in order to settle completely the above conjecture new ideas rather than just a long computation are needed. Classical and modular approaches to exponential diophantine. In this study, we give three identities concerning fibonacci and lucas numbers. Since these rules a re proved i n, we omit their proofs. On some diophantine equations journal of inequalities. Siksek, classical and modular approaches to exponential diophantine equations i, fibonacci and lucas perfect powers, annals of mathematics. Moreover, we obtain the set of positive integer solutions of some specific pell equations in terms of the integer sequences mentioned in the text. However, it is still known that there are no prime solutions when n pelllucas, and balancing numbers and the relationships between them.
Solutions of some diophantine equations in terms of. By using the generalized fibonacci and lucas sequences, we obtain all integer solutions of some diophantine equations such as x2kxyy2. Three identities concerning fibonacci and lucas numbers. Pdf solutions of some diophantine equations using generalized. God made the integers, all else is the work of man. Diophantine equations with products of consecutive terms. In this paper, we are interested in fibonacci and lucas numbers which are sum of three perfect powers of some prescribed distinct bases. 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. In this paper, we derive some identities on pell, pelllucas, and balancing numbers and the relationships between them. In 12, the crucial point of the proof for the case of the fibonacci sequence was the existence of a factorizationof f n. Fibonacci numbers, lucas numbers, fibonacci matrix, diophantine equation. There are some diophantine equations involving the fibonomial coe cients and fibonacci numbers. We consider the sequences u n and v n which are the generalizations of fibonacci and lucas sequences, respectively. For example, marques 5 investigated the solutions of the fibonacci version of the brocardramanujan diophantine equation and showed that the diophantine equation 1.
Let ln be the nth term of the lucas sequence defined by. Fibonacci and lucas numbers with applications thomas koshy. Positive integer solutions of some diophantine equations. Diophantine equations, continuity fractions, binary recurrent sequences, fibonacci numbers, pell numbers, lower bounds for linear forms in logarithms. There are many diophantine equations that have all solutions given in terms of particular lucas sequences as fibonacci and lucas numbers e. On some diophantine equations, journal of inequalities and. Fibonacci numbers for negative subscripts are defined as f. In the first part of this paper, some problems related to prime numbers and the role of diophantine equations in design theory is discussed. Integer solutions of some diophantine equations via fibonacci and lucas numbers bahar demirturk and refik keskin department of mathematics sakarya university. The fibonacci version of the brocardramanujan diophantine equation diego marques abstract. Keywords fibonacci numbers lucas numbers generalized fibonacci numbers generalized lucas numbers diophantine equations. The sequences of fibonacci numbers and lucas numbers.
Lucas and fibonacci numbers and some diophantine equations, proc. Balancing numbers were first considered by behera and panda in when they considered the integer solutions of the diophantine equation for some positive. This function can be implemented in software mathematica. The fibonacci version of the brocardramanujan diophantine. Solve integer equations, determine remainders of powers, and much more with the power of modular arithmetic. Pdf in this study, we deal with some diophantine equations. The lucas numbers satisfy remarkable difference equations, in some cases best expressed using fibonacci numbers, of which representative examples are the following. Schoenfeld, approximate formulas for some functions of prime numbers, illinois j. Using these equations, we also explore all integer solutions of some other diophantine equations. The elementary properties of these numbers are easily established. 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. Such results might have some interest in their own and could be useful for other diophantine questions involving fibonacci numbers. Lucas and fibonacci number ans d some diophantine equations by j.
On the diophantine equation z n k n involving the order. Florian luca diophantine sets of fibonacci numbers. On diophantine equations related to order of appearance in. Fibonacci and lucas perfect powers by yann bugeaud, maurice mignotte, and samir siksek abstract this is the. Solutions to some problems related to diophantine equation.
The fibonacci version of the brocardramanujan diophantine equation diego marques. Siksek, classical and modular approaches to exponential diophantine equations i. Fibonacci numbers, lucas numbers, fibonacci matrix, diophantine. Fibonacci and lucas perfect powers yann bugeaud, maurice mignotte, samir siksek abstract. 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. In this study, we give some new identities concerning fibonacci and lucas numbers. Ams mathematics of computation american mathematical society. Here, the case of the lucas numbers is more complicated since such a factorization of l. Heptagonal numbers in the associated pell sequence and diophantine equations x25x. The contribution of fibonacci and lucas numbers to a. Fibonacci and lucas numbers, factorials, diophantine equations. In this note, we prove that the fibonacci version of the brocardramanujan diophantine equation n.
Solutions of some of the above equations are related to the fibonacci numbers. Duman and tuglu 2015, kitayama and shiomi 2017, remis 2011, wang and wang 2017, ye and zhang 2017. Solutions of some quadratic diophantine equations core. 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. Pdf we study the problem of finding all integer solutions of the diophantine equations x2. Positive integer solutions of some secondorder diophantine. However, it is still known that there are no prime solutions when n 0 and ei. On diophantine equations involving lucas sequences. By using the generalized fibonacci and lucas sequences, we obtain all.
The tenth problem of hilbert was the problem of existence of an algorithm for deciding solvability of diophantine equations. Then, we give all positive integer solutions of some diophantine equations in terms of these new sequences. Pdf we study the problem of finding all integer solutions of the diophantine. 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. On some diophantine equations journal of inequalities and. Larger families of solutions to some diophantine equations.
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