On Fibonacci and Lucas sequences modulo a prime and primality testing

Andrica, Dorin; Crişan, Vlad; Al-Thukair, Fawzi

doi:10.1016/j.ajmsc.2017.06.002

Cited by 10 publications

(11 citation statements)

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“…In Section 2 we review the notion of Fibonacci pseudoprime of level k, and propose a counterpart defined for Lucas sequences. We also disprove a statement formulated in [14] for Fibonacci numbers, which shows that the relationship between the pseudoprimes of different levels is not trivial. In Section 3 we define the generalized Lucas and Pell-Lucas pseudoprimality of level k, which involves the Jacobi symbol.…”

Section: Propositionsupporting

confidence: 53%

“…In this section we present the Fibonacci pseudoprimes of level k and give a counterexample to a result from [14], about the connection between the sets of pseudoprimes on different levels. We then define the Lucas pseudoprimes of level k, for which we also explore connections between the pseudoprimes on different levels.…”

Section: Fibonacci and Lucas Pseudoprimes Of Level Kmentioning

confidence: 99%

“…A composite number n is called a Fibonacci pseudoprime if n | F n−( n 5 ) . The even Fibonacci pseudoprimes are indexed as A141137 in the OEIS [15] [14], the authors introduced the following notion. Let k be a fixed positive integer.…”

Section: Fibonacci Pseudoprimes Of Level Kmentioning

confidence: 99%

“…Notice that for k = 1 we obtain the classical Fibonacci pseudoprimes. We now state a corrected version of Proposition 1 in [14], and then discuss why the original version does not hold. Proposition 2.…”

Section: Fibonacci Pseudoprimes Of Level Kmentioning

confidence: 99%

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On Generalized Lucas Pseudoprimality of Level k

Andrica

Bagdasar

2021

Mathematics

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We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels k+ and k− and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences.

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Section: Propositionsupporting

confidence: 53%

Section: Fibonacci and Lucas Pseudoprimes Of Level Kmentioning

confidence: 99%

Section: Fibonacci Pseudoprimes Of Level Kmentioning

confidence: 99%

Section: Fibonacci Pseudoprimes Of Level Kmentioning

confidence: 99%

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On Generalized Lucas Pseudoprimality of Level k

Andrica

Bagdasar

2021

Mathematics

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“…) . (20) where k,k 1 and k 2 are positive integers not less than 1 and the set of values of r 1 ,r 2 is {4,1,2}.…”

Section: Integer Factorization Algorithm Using the Pisano Periodmentioning

confidence: 99%

The Integer Factorization Algorithm With Pisano Period

Cai

Gong

2019

IEEE Access

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Large integer factorization is one of the basic issues in number theory and is the subject of this paper. Our research shows that the Pisano period of the product of two prime numbers (or an integer multiple of it) can be derived from the two prime numbers themselves and their product, and we can therefore decompose the two prime numbers by means of the Pisano period of their product. We reduce the computational complexity of modulo operation through the ''fast Fibonacci modulo algorithm'' and design a stochastic algorithm for finding the Pisano periods of large integers. The Pisano period factorization method, which is proved to be slightly better than the quadratic sieve method and the elliptic curve method, consumes as much time as Fermat's method, the continued fractional factorization method and the Pollard p-1 method on small integer factorization cases. When factoring super-large integers, the Pisano period factorization method has shown as strong performance as subexponential complexity methods; thus, this method demonstrates a certain practicability. We suggest that this paper may provide a completely new idea in the area of integer factorization problems.

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Arithmetic and Trigonometric Properties of Some Classical Recurrent Sequences

Andrica

Bagdasar

2020

Problem Books in Mathematics

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On Fibonacci and Lucas sequences modulo a prime and primality testing

Cited by 10 publications

References 1 publication

On Generalized Lucas Pseudoprimality of Level k

On Generalized Lucas Pseudoprimality of Level k

The Integer Factorization Algorithm With Pisano Period

Arithmetic and Trigonometric Properties of Some Classical Recurrent Sequences

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