# Fibonacci Sequence And Lucas Sequence Modulo

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Calculate Huge Fibonacci number modulo M in C++. Ask Question Asked 3 years, 4 months. Note that you need to take the number and half it repeatedly, subtracting 1 where needed, to figure out the sequence to go the other way. Then you work your way back up using the above to double. Calculating a huge fibonacci number modulo m in Python. 2.

Recall that the sequence of Fibonacci numbers, reduced modulo an integer m ≥ 2, is purely periodic. Renault [4] gave a nice discussion of the properties of the reduced Fibonacci sequence, including a list of the periods for each m. We will be interested in the period when reducing modulo a Lucas number.

Note that if the Lucas sequence modulo m contains a complete set of residues then the Fibonacci sequence must also. (If the Lucas sequence contains 0 followed by d, then it continues as d times the Fibonacci sequence.) As 5 does not work this rules out all m divisible by 5, checking 2,4,6,7,14 by hand only leaves the powers of 3 undecided.

Z[’] AND THE FIBONACCI SEQUENCE MODULO n SAMIN RIASAT Abstract. It has long been known that the Fibonacci sequence modulo n is periodic for any integer n > 1. In this paper we present an elementary approach of proving properties of this period by working on Z[’] and also deduce some new results.

It is shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic. Also if is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to modulo. It is found that order of cyclic group generated with matrix is equal to the period of these sequences.

Splitting fields and periods of Fibonacci sequences modulo primes. Math. Mag., Volume 85, Number 2, April 2012, 130-135. 130 MATHEMATICS MAGAZINE REFERENCES 1.Ward Cheney and Will Light, A Course in Approximation Theory, American Mathematical Society, Provi-dence, RI, 2009

DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k. , title = {DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k}, year = {}} Share. OpenURL. Abstract. Dedicated to Peter Shiue on the occasion of his 70th birthday Abstract. Let F0 = 0,F1 = 1, and Fn = Fn−1 +Fn−2 for n ≥ 2 denote the sequence F of Fibonacci numbers.

Note that if the Lucas sequence modulo m contains a complete set of residues then the Fibonacci sequence must also. (If the Lucas sequence contains 0 followed by d, then it continues as d times the Fibonacci sequence.) As 5 does not work this rules out all m divisible by 5, checking 2,4,6,7,14 by hand only leaves the powers of 3 undecided.

CiteSeerX – Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Dedicated to Peter Shiue on the occasion of his 70th birthday Abstract. Let F0 = 0,F1 = 1, and Fn = Fn−1 +Fn−2 for n ≥ 2 denote the sequence F of Fibonacci numbers. For any modulus m ≥ 2 and residue b (modm), denote by vF(m,b) the number of occurrences of b as a residue in one (shortest) period of F modulo m.

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A Fibonacci-like Sequence of Composite Numbers DONALD E. KNUTH* Stanford University Stanford, CA 94305 Ronald L. Graham [1] found relatively prime integers a and b such that the sequence

Definition: By F(mod m) we mean the sequence of least nonnegative residues of the terms of the Fibonacci sequence taken modulo m. This is a bi-infinite sequence in that given any two consecutive terms, we can find the terms preceeding and following those terms. For example, I am using F(mod 5) as the horizontal rule above.

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Periodic generalized Fibonacci sequences modulo m have been studied in a number of papers (see e.g. [9, 11,19,21]). In this section, we consider the periodic generalized Fibonacci sequence.

This paper is concerned with periods of Biperiodic Fibonacci and Biperiodic Lucas sequences taken as modulo prime and prime power. By using Fermat’s little theorem, quadratic reciprocity, many results are obtained. 1. Introduction. Fibonacci sequence and Lucas sequence are well-known sequences among integer sequences.

They are transformed using Fibonacci congruences and we will get by this way sequencesn. These se-x quences xn admit the IID model for correct model. Keywords: Fibonacci Sequence, Linear Congruence, Random Numbers, Dependence, Correct Models. 1. Introduction. In this paper, we present a new method using Fibonacci sequences to obtain real IID.

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3Step Fibonacci series modulo m. Article. We can see applications of the Fibonacci sequence in group theory in [4, 6] and also see some generalized Fibonacci and Lucas sequences in.

The thing is that if we have a look at the Fibonacci sequence after applying to each element a modulo operator, we would notice that we always get periodic sequences. Modulo 2 leads to a infinite repetition of three elements: 0, 1, 1 Modulo 3 to eight elements: 0, 1, 1, 2, 0, 2, 2, 1 Modulo 10 to sixty, et cetera.

Definition: By F(mod m) we mean the sequence of least nonnegative residues of the terms of the Fibonacci sequence taken modulo m. This is a bi-infinite sequence in that given any two consecutive terms, we can find the terms preceeding and following those terms. For example, I am using F(mod 5) as the horizontal rule above.

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