141 research outputs found
S-parts of terms of integer linear recurrence sequences
^Δ holds for n > n_0 . Our proof is ineffective in the sense that it does not give an explicit value for n_0. Under various assumptions on (u_n)_{nâ„0}, we also give effective, but weaker, upper bounds for [u_n]_S of the form ^^Let S = {q1 , . . . , qs } be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q1^ r1 . . . qs^rs M, where r1 , . . . , rs are non-negative integers and M is an integer relatively prime to q1 . . . qs. We define the S-part [m]_S of m by [m]_S := q1^r1 . . . qs^rs.Let (u_n )_{nâ„0} be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every Δ > 0, there exists an integer n_0 such that [u_n ]_S †(u_n )Article / Letter to editorMathematisch Instituu
S-parts of terms of integer linear recurrence sequences
^Δ holds for n > n_0 . Our proof is ineffective in the sense that it does not give an explicit value for n_0. Under various assumptions on (u_n)_{nâ„0}, we also give effective, but weaker, upper bounds for [u_n]_S of the form ^^Let S = {q1 , . . . , qs } be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q1^ r1 . . . qs^rs M, where r1 , . . . , rs are non-negative integers and M is an integer relatively prime to q1 . . . qs. We define the S-part [m]_S of m by [m]_S := q1^r1 . . . qs^rs.Let (u_n )_{nâ„0} be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every Δ > 0, there exists an integer n_0 such that [u_n ]_S †(u_n )Article / Letter to editorMathematisch Instituu
S-parts of terms of integer linear recurrence sequences
Let S = {q1 , . . . , qs } be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q1^ r1 . . . qs^rs M, where r1 , . . . , rs are non-negative integers and M is an integer relatively prime to q1 . . . qs. We define the S-part [m]_S of m by [m]_S := q1^r1 . . . qs^rs.Let (u_n )_{nâ„0} be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every Δ > 0, there exists an integer n_0 such that [u_n ]_S †|u_n |^Δ holds for n > n_0 . Our proof is ineffective in the sense that it does not give an explicit value for n_0. Under various assumptions on (u_n)_{nâ„0}, we also give effective, but weaker, upper bounds for [u_n]_S of the form |u_n|^{1âc} , where c is positive and depends only on (u_n)_{nâ„0} and S.Number theory, Algebra and Geometr
There are no multiply-perfect Fibonacci numbers
Here, we show that no Fibonacci number (larger than 1) divides the sum of its divisors
Extending Elliptic Curve Chabauty to higher genus curves
We give a generalization of the method of "Elliptic Curve Chabauty" to higher
genus curves and their Jacobians. This method can sometimes be used in
conjunction with covering techniques and a modified version of the Mordell-Weil
sieve to provide a complete solution to the problem of determining the set of
rational points of an algebraic curve .Comment: 24 page
Asymptotic diophantine approximation:the multiplicative case
Let and be irrational real numbers and 0<\F<1/30. We prove
a precise estimate for the number of positive integers that satisfy
\|q\alpha\|\cdot\|q\beta\|<\F. If we choose \F as a function of we get
asymptotics as gets large, provided \F Q grows quickly enough in terms of
the (multiplicative) Diophantine type of , e.g., if
is a counterexample to Littlewood's conjecture then we only
need that \F Q tends to infinity. Our result yields a new upper bound on sums
of reciprocals of products of fractional parts, and sheds some light on a
recent question of L\^{e} and Vaaler.Comment: To appear in Ramanujan Journa
Rational approximation to algebraic varieties and a new exponent of simultaneous approximation
The Critical Exponent is Computable for Automatic Sequences
The critical exponent of an infinite word is defined to be the supremum of
the exponent of each of its factors. For k-automatic sequences, we show that
this critical exponent is always either a rational number or infinite, and its
value is computable. Our results also apply to variants of the critical
exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni
and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes
or recovers previous results of Krieger and others, and is applicable to other
situations; e.g., the computation of the optimal recurrence constant for a
linearly recurrent k-automatic sequence.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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