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Register a new userOn a problem of Arnold: the average multiplicative order of a given integer
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For g,n coprime integers, let l_g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l_g(n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l_g(p) as p <= x ranges over primes.
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We obtain explicit lower bounds on multiplicative order of elements that have more general form than finite field Gauss period. In a partial case of Gauss period this bound improves the previous bound of O.Ahmadi, I.E.Shparlinski and J.F.Voloch
Let $G$ be a graph and $tau$ be an assignment of nonnegative integer thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a $tau$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, ldots, D_k$ such that $D_0=D$ and for any $iin {0, ldots, k-1}$, each vertex $v$ in $D_{i+1}$ has at least $tau(v)$ neighbors in $D_0cup ldots cup D_i$. Denote the size of smallest $tau$-dynamic monopoly by $dyn_{tau}(G)$ and the average of thresholds in $tau$ by $overline{tau}$. We show that the values of $dyn_{tau}(G)$ over all assignments $tau$ with the same average threshold is a continuous set of integers. For any positive number $t$, denote the maximum $dyn_{tau}(G)$ taken over all threshold assignments $tau$ with $overline{tau}leq t$, by $Ldyn_t(G)$. In fact, $Ldyn_t(G)$ shows the worst-case value of a dynamic monopoly when the average threshold is a given number $t$. We investigate under what conditions on $t$, there exists an upper bound for $Ldyn_{t}(G)$ of the form $c|G|$, where $c<1$. Next, we show that $Ldyn_t(G)$ is coNP-hard for planar graphs but has polynomial-time solution for forests.
Write $mathrm{ord}_p(cdot)$ for the multiplicative order in $mathbb{F}_p^{times}$. Recently, Matthew Just and the second author investigated the problem of classifying pairs $alpha, beta in mathbb{Q}^{times}setminus{pm 1}$ for which $mathrm{ord}_p(alpha) > mathrm{ord}_p(beta)$ holds for infinitely many primes $p$. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for $alpha,beta$ to be order-dominant. Via the large sieve, we show that almost all integer pairs $alpha,beta$ satisfy our condition, with a power savings on the size of the exceptional set.
Using the following $_4F_3$ transformation formula $$ sum_{k=0}^{n}{-x-1choose k}^2{xchoose n-k}^2=sum_{k=0}^{n}{n+kchoose 2k}{2kchoose k}^2{x+kchoose 2k}, $$ which can be proved by Zeilbergers algorithm, we confirm some special cases of a recent conjecture of Z.-W. Sun on integer-valued polynomials.
In this note, I study a comparison map between a motivic and {e}tale cohomology group of an elliptic curve over $mathbb{Q}$ just outside the range of Voevodskys isomorphism theorem. I show that the property of an appropriate version of the map being an isomorphism is equivalent to certain arithmetical properties of the elliptic curve.
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