442 research outputs found

    Distribution of integral values for the ratio of two linear recurrences

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    Let FF and GG be linear recurrences over a number field K\mathbb{K}, and let R\mathfrak{R} be a finitely generated subring of K\mathbb{K}. Furthermore, let N\mathcal{N} be the set of positive integers nn such that G(n)0G(n) \neq 0 and F(n)/G(n)RF(n) / G(n) \in \mathfrak{R}. Under mild hypothesis, Corvaja and Zannier proved that N\mathcal{N} has zero asymptotic density. We prove that #(N[1,x])x(loglogx/logx)h\#(\mathcal{N} \cap [1, x]) \ll x \cdot (\log\log x / \log x)^h for all x3x \geq 3, where hh is a positive integer that can be computed in terms of FF and GG. Assuming the Hardy-Littlewood kk-tuple conjecture, our result is optimal except for the term loglogx\log \log x

    On the sum of digits of the factorial

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    Let b > 1 be an integer and denote by s_b(m) the sum of the digits of the positive integer m when is written in base b. We prove that s_b(n!) > C_b log n log log log n for each integer n > e, where C_b is a positive constant depending only on b. This improves of a factor log log log n a previous lower bound for s_b(n!) given by Luca. We prove also the same inequality but with n! replaced by the least common multiple of 1,2,...,n.Comment: 4 page

    Covering an arithmetic progression with geometric progressions and vice versa

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    We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric progressions to cover the first n terms of A. A similar result is presented, with the role of arithmetic and geometric progressions reversed.Comment: 4 page

    A note on primes in certain residue classes

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    Given positive integers a1,,aka_1,\ldots,a_k, we prove that the set of primes pp such that p≢1modaip \not\equiv 1 \bmod{a_i} for i=1,,ki=1,\ldots,k admits asymptotic density relative to the set of all primes which is at least i=1k(11φ(ai))\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right), where φ\varphi is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer nn such that n≢0modain \not\equiv 0 \bmod a_i for i=1,,ki=1,\ldots,k admits asymptotic density which is at least i=1k(11ai)\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)

    On the greatest common divisor of nn and the nnth Fibonacci number

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    Let A\mathcal{A} be the set of all integers of the form gcd(n,Fn)\gcd(n, F_n), where nn is a positive integer and FnF_n denotes the nnth Fibonacci number. We prove that #(A[1,x])x/logx\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x for all x2x \geq 2, and that A\mathcal{A} has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer nn, an explicit formula for the density of primes pp such that nn divides the rank of appearance of pp, that is, the smallest positive integer kk such that pp divides FkF_k

    A coprimality condition on consecutive values of polynomials

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    Let fZ[X]f\in\mathbb{Z}[X] be quadratic or cubic polynomial. We prove that there exists an integer Gf2G_f\geq 2 such that for every integer kGfk\geq G_f one can find infinitely many integers n0n\geq 0 with the property that none of f(n+1),f(n+2),,f(n+k)f(n+1),f(n+2),\dots,f(n+k) is coprime to all the others. This extends previous results on linear polynomials and, in particular, on consecutive integers

    The density of numbers nn having a prescribed G.C.D. with the nnth Fibonacci number

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    For each positive integer kk, let Ak\mathscr{A}_k be the set of all positive integers nn such that gcd(n,Fn)=k\gcd(n, F_n) = k, where FnF_n denotes the nnth Fibonacci number. We prove that the asymptotic density of Ak\mathscr{A}_k exists and is equal to d=1μ(d)lcm(dk,z(dk))\sum_{d = 1}^\infty \frac{\mu(d)}{\operatorname{lcm}(dk, z(dk))} where μ\mu is the M\"obius function and z(m)z(m) denotes the least positive integer nn such that mm divides FnF_n. We also give an effective criterion to establish when the asymptotic density of Ak\mathscr{A}_k is zero and we show that this is the case if and only if Ak\mathscr{A}_k is empty
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