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Terms of Lucas sequences having a large smooth divisor

2022
*
Canadian mathematical bulletin
*

We show that the 𝐾 𝑛-

doi:10.4153/s0008439522000248
fatcat:cug5xctfzfbltlyj25gw2wd4ei
*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. ...##
###
ALGEBRAIC DIVISIBILITY SEQUENCES OVER FUNCTION FIELDS

2012
*
Journal of the Australian Mathematical Society
*

We also prove that an elliptic divisibility

doi:10.1017/s1446788712000092
fatcat:vx3muzpsx5ejbhwid7roako6ny
*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. ...##
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Primitive Divisors, Dynamical Zsigmondy Sets, and Vojta's Conjecture
[article]

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. ...

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

2007
*
arXiv
*
pre-print

We survey some results about

arXiv:math/0701234v1
fatcat:odgwysdmf5ci3n52gaesj4cgfa
*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*. ...##
###
On numbers n dividing the nth term of a linear recurrence
[article]

2011
*
arXiv
*
pre-print

Here, we give upper and lower bounds on the count

arXiv:1010.4544v2
fatcat:a2qhcumkjbce3bonpvwfegfj2m
*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). ...##
###
Smooth values of some quadratic polynomials
[article]

2010
*
arXiv
*
pre-print

In this paper, using

arXiv:1005.1535v1
fatcat:sn4wjf4kj5go5my263kw7pfj5m
*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. ...##
###
Smooth values of some quadratic polynomials

2010
*
Glasnik Matematicki - Serija III
*

In this paper, using

doi:10.3336/gm.45.2.04
fatcat:7ydrzwhxpfdupjlfbo2kkuuil4
*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. ...##
###
Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function

2006
*
Indagationes mathematicae
*

The same result holds for Mersenne numbers 2 n -1 and for one more general class

doi:10.1016/s0019-3577(06)81037-2
fatcat:h5o3islsm5gfbaqvggeqoqu5au
*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. ...##
###
Davenport constant for finite abelian groups

2008
*
Indagationes mathematicae
*

TI]deman ABSTRACT For

doi:10.1016/s0019-3577(08)00006-2
fatcat:dvmfjz4vt5dufiwki75a5b6z4i
*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. ...##
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Divisibility, Smoothness and Cryptographic Applications
[article]

2009
*
arXiv
*
pre-print

We present various properties

arXiv:0810.2067v3
fatcat:2i7q2fbwijcrjkvirs53dawcfm
*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 ...##
###
How Smooth Is $\phi(2^n+3)$?

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

doi:10.1216/rmjm/1181069806
fatcat:aenrtvpbnfb5rafir25bislqgm
*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. ...##
###
Arithmetic properties of the Ramanujan function
[article]

2006
*
arXiv
*
pre-print

We study some arithmetic properties

arXiv:math/0607591v1
fatcat:r2dhkil56na3jpn5hn2vkagj7u
*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 ...##
###
On numbers n dividing the nth term of a linear recurrence

2012
*
Proceedings of the Edinburgh Mathematical Society
*

Here, we give upper and lower bounds on the count

doi:10.1017/s0013091510001355
fatcat:kmrx5irbkfc27latisrcfer5fq
*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). ...##
###
The reciprocal sum of divisors of Mersenne numbers

2020
*
Acta Arithmetica
*

Whereas all but finitely many primes divide some

doi:10.4064/aa200602-11-9
fatcat:ml7mk354yngv5fx7kwp64ii7nm
*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. ...##
###
The reciprocal sum of divisors of Mersenne numbers
[article]

2020
*
arXiv
*
pre-print

We investigate various questions concerning the reciprocal sum

arXiv:2006.02373v2
fatcat:2ztksfj3qbdpblmolj2hqaj7wq
*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. ...
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