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Terms of Lucas sequences having a large smooth divisor

Nikhil Balaji, Florian Luca
2022 Canadian mathematical bulletin  
We show that the 𝐾 𝑛-smooth part of 𝑎 𝑛 − 1 for an integer 𝑎 > 1 is 𝑎 𝑜(𝑛) for most positive integers 𝑛.  ...  Our results are easily extendable to all Lucas sequences, in particular, the sequence of Fibonacci numbers. To start we recall the famous 𝐴𝐵𝐶-conjecture.  ...  We then have 𝑎 𝑛 ≪ 𝜀 (𝑎 • rad(𝑠)𝑡)) 1+𝜀 ≪ 𝑝≤𝐾 𝑛 𝑝 | 𝑎 𝑛 −1 𝑝 1+𝜀 (𝑎/𝑐) 𝑛(1+𝜀 Terms of Lucas sequences having a smooth divisor 3 We may of course assume that 1 < 𝑐 < 𝑎.  ... 
doi:10.4153/s0008439522000248 fatcat:cug5xctfzfbltlyj25gw2wd4ei

ALGEBRAIC DIVISIBILITY SEQUENCES OVER FUNCTION FIELDS

PATRICK INGRAM, VALÉRY MAHÉ, JOSEPH H. SILVERMAN, KATHERINE E. STANGE, MARCO STRENG
2012 Journal of the Australian Mathematical Society  
We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.  ...  We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields.  ...  Graham is unfortunately no longer with us, but his ideas suffuse this work, and we take this opportunity to remember and appreciate his life as a valued colleague and friend.  ... 
doi:10.1017/s1446788712000092 fatcat:vx3muzpsx5ejbhwid7roako6ny

Primitive Divisors, Dynamical Zsigmondy Sets, and Vojta's Conjecture [article]

Joseph H. Silverman
2012 arXiv   pre-print
A primitive prime divisor of an element a_n of a sequence (a_1,a_2,a_3,...) is a prime P that divides a_n, but does not divide a_m for all m < n.  ...  Let f : X --> X be a self-morphism of a variety, let D be an effective divisor on X, and let P be a point of X, all defined over the algebraic closure of Q.  ...  The study of primitive divisors in Lucas sequence was completed in 2001 by Bilu, Hanrot, and Voutier [5] , who proved that a Lucas sequence has primitive divisors for each term with n > 30.  ... 
arXiv:1209.3491v1 fatcat:6n7bszudljayxgnrdlrrki4xpq

On primitive divisors of n^2+b [article]

Graham Everest, Glyn Harman
2007 arXiv   pre-print
We survey some results about divisors of this sequence as well as provide upper and lower growth estimates for the number of terms which have a primitive divisor.  ...  We study primitive divisors of terms of the sequence P_n=n^2+b, for a fixed integer b which is not a negative square.  ...  Primitive divisors of n 2 + b. Theorem 1.2. Infinitely many terms of the sequence n 2 + b do not have a primitive divisor.  ... 
arXiv:math/0701234v1 fatcat:odgwysdmf5ci3n52gaesj4cgfa

On numbers n dividing the nth term of a linear recurrence [article]

Juan Jose Alba Gonzalez, Florian Luca, Carl Pomerance, Igor Shparlinski
2011 arXiv   pre-print
Here, we give upper and lower bounds on the count of positive integers n< x dividing the nth term of a nondegenerate linearly recurrent sequence with simple roots.  ...  During the preparation of this paper, F. L. was supported in part by grants SEP-CONACyT 79685 and PAPIIT 100508, C. P. was supported in part by NSF grant DMS-1001180 and I.  ...  For Lucas sequences with a 2 = ±1, we also have a rather strong lower bound on #N u (x).  ... 
arXiv:1010.4544v2 fatcat:a2qhcumkjbce3bonpvwfegfj2m

Smooth values of some quadratic polynomials [article]

Filip Najman
2010 arXiv   pre-print
In this paper, using a method of Luca and the author, we find all values x such that the quadratic polynomials x^2+1, x^2+4, x^2+2 and x^2-2 are 200-smooth and all values x such that the quadratic polynomial  ...  x^2-4 is 100-smooth.  ...  Many thanks go to Florian Luca for pointing out that this method will work for x 2 ± 2. I am also grateful to Andrej Dujella and the referee for many helpful comments.  ... 
arXiv:1005.1535v1 fatcat:sn4wjf4kj5go5my263kw7pfj5m

Smooth values of some quadratic polynomials

Filip Najman
2010 Glasnik Matematicki - Serija III  
In this paper, using a method of Luca and the author, we find all values x such that the quadratic polynomials x 2 + 1, x 2 + 4, x 2 + 2 and x 2 − 2 are 200-smooth and all values x such that the quadratic  ...  polynomial x 2 − 4 is 100-smooth.  ...  Many thanks go to Florian Luca for pointing out that this method will work for x 2 ± 2. I am also grateful to Andrej Dujella and the referee for many helpful comments.  ... 
doi:10.3336/gm.45.2.04 fatcat:7ydrzwhxpfdupjlfbo2kkuuil4

Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function

Florian Luca, Igor E. Shparlinski
2006 Indagationes mathematicae  
The same result holds for Mersenne numbers 2 n -1 and for one more general class of Lucas sequences.  ...  A slight modification of our method also leads to similar results for polynomial sequences f(n), where f E Z[XI.  ...  ACKNOWLEDGEMENTS The authors are grateful to the referee whose suggestions have lead to a shorter proof and sharper conclusion of Theorem 4.  ... 
doi:10.1016/s0019-3577(06)81037-2 fatcat:h5o3islsm5gfbaqvggeqoqu5au

Davenport constant for finite abelian groups

Emre Alkan
2008 Indagationes mathematicae  
TI]deman ABSTRACT For a fimte abelian group G. we mvestlgate the length of a sequence of elements of G that IS guaranteed to have a subsequence with product Identity of G.  ...  TI]deman ABSTRACT For a fimte abelian group G. we mvestlgate the length of a sequence of elements of G that IS guaranteed to have a subsequence with product Identity of G.  ...  , Luca and Shparlinski [3] and the remark following the proof of Theorem 4.  ... 
doi:10.1016/s0019-3577(08)00006-2 fatcat:dvmfjz4vt5dufiwki75a5b6z4i

Divisibility, Smoothness and Cryptographic Applications [article]

David Naccache, Igor E. Shparlinski
2009 arXiv   pre-print
We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences.  ...  This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play a crucial role in information theory, signal processing and cryptography.  ...  Large Smooth Divisors It also natural to ask how often integers are expected to have a large smooth divisor; or, from a more quantitative perspective, explore the behavior of: Θ(x, y, z) = #{n ≤ x : ∃d  ... 
arXiv:0810.2067v3 fatcat:2i7q2fbwijcrjkvirs53dawcfm

How Smooth Is $\phi(2^n+3)$?

Florian Luca
2004 Rocky Mountain Journal of Mathematics  
In this paper, we show, among other things, that P (φ(2 n + 3)) tends to infinity with n on a set of n of asymptotic density 1.  ...  For any integer n let P (n) denote the largest prime divisor of n with the convention that P (±1) = P (0) = 1. We also let φ(n) denote the Euler function of n.  ...  I would like to thank the referee for comments and suggestions which improved the presentation of this paper.  ... 
doi:10.1216/rmjm/1181069806 fatcat:aenrtvpbnfb5rafir25bislqgm

Arithmetic properties of the Ramanujan function [article]

Florian Luca, Igor E Shparlinski
2006 arXiv   pre-print
We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisor P(τ(n)) and the number of distinct prime divisors ω(τ(n)) of τ(n) for various sequences of n.  ...  Acknowledgements During the preparation of this paper, the first author was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and the second author was supported in part by ARC grant DP0211459  ...  By the primitive divisor theorem for Lucas sequences which claims that each sufficiently large term u r has at least one new prime divisor (see [2] for the most general form of this assertion), we conclude  ... 
arXiv:math/0607591v1 fatcat:r2dhkil56na3jpn5hn2vkagj7u

On numbers n dividing the nth term of a linear recurrence

Juan José Alba González, Florian Luca, Carl Pomerance, Igor E. Shparlinski
2012 Proceedings of the Edinburgh Mathematical Society  
Here, we give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a nondegenerate linearly recurrent sequence with simple roots.  ...  During the preparation of this paper, F. L. was supported in part by grants SEP-CONACyT 79685 and PAPIIT 100508, C. P. was supported in part by NSF grant DMS-1001180 and I.  ...  For Lucas sequences with a 2 = ±1, we also have a rather strong lower bound on #N u (x).  ... 
doi:10.1017/s0013091510001355 fatcat:kmrx5irbkfc27latisrcfer5fq

The reciprocal sum of divisors of Mersenne numbers

Zebediah Engberg, Paul Pollack
2020 Acta Arithmetica  
Whereas all but finitely many primes divide some term in the U n sequence, with finitely many exceptions a prime p divides some term in the V n sequence if and only if z(p) is defined and even.  ...  In the proof of Theorem 5.4, q will be chosen as a divisor of a /a, and so the prime power q e will have q > q 0 . An example and a non-example.  ... 
doi:10.4064/aa200602-11-9 fatcat:ml7mk354yngv5fx7kwp64ii7nm

The reciprocal sum of divisors of Mersenne numbers [article]

Zebediah Engberg, Paul Pollack
2020 arXiv   pre-print
We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers 2^n-1.  ...  This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman.  ...  Whereas all but finitely many primes divide some term in the U n sequence, with finitely many exceptions a prime p divides some term in the V n sequence if and only if z(p) is defined and even.  ... 
arXiv:2006.02373v2 fatcat:2ztksfj3qbdpblmolj2hqaj7wq
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