S-parts of terms of integer linear recurrence sequences

^ε 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)_{n≥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 )_{n≥0} be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every ε > 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

^ε 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)_{n≥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 )_{n≥0} be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every ε > 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 )_{n≥0} be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every ε > 0, there exists an integer n_0 such that [u_n ]_S ≤ |u_n |^ε 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)_{n≥0}, we also give effective, but weaker, upper bounds for [u_n]_S of the form |u_n|^{1−c} , where c is positive and depends only on (u_n)_{n≥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 $Y$.Comment: 24 page

Asymptotic diophantine approximation:the multiplicative case

Let $\alpha$ and $\beta$ be irrational real numbers and 0<\F<1/30. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy \|q\alpha\|\cdot\|q\beta\|<\F. If we choose \F as a function of $Q$ we get asymptotics as $Q$ gets large, provided \F Q grows quickly enough in terms of the (multiplicative) Diophantine type of $(\alpha,\beta)$, e.g., if $(\alpha,\beta)$ 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

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