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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.
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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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In 1947 Mills proved that there exists a constant $A$ such that $lfloor A^{3^n} rfloor$ is a prime for every positive integer $n$. Determining $A$ requires determining an effective Hoheisel type result on the primes in short intervals - though most books ignore this difficulty. Under the Riemann Hypothesis, we show that there exists at least one prime between every pair of consecutive cubes and determine (given RH) that the least possible value of Mills constant $A$ does begin with 1.3063778838. We calculate this value to 6850 decimal places by determining the associated primes to over 6000 digits and probable primes (PRPs) to over 60000 digits. We also apply the Cramer-Granville Conjecture to Honakers problem in a related context.
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