اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدVariations on a theme of Schinzel and Wojcik
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Schinzel and Wojcik have shown that if $alpha, beta$ are rational numbers not $0$ or $pm 1$, then $mathrm{ord}_p(alpha)=mathrm{ord}_p(beta)$ for infinitely many primes $p$, where $mathrm{ord}_p(cdot)$ denotes the order in $mathbb{F}_p^{times}$. We begin by asking: When are there infinitely many primes $p$ with $mathrm{ord}_p(alpha) > mathrm{ord}_p(beta)$? We write down several families of pairs $alpha,beta$ for which we can prove this to be the case. In particular, we show this happens for 100% of pairs $A,2$, as $A$ runs through the positive integers. We end on a different note, proving a version of Schinzel and W{o}jciks theorem for the integers of an imaginary quadratic field $K$: If $alpha, beta in mathcal{O}_K$ are nonzero and neither is a root of unity, then there are infinitely many maximal ideals $P$ of $mathcal{O}_K$ for which $mathrm{ord}_P(alpha) = mathrm{ord}_P(beta)$.
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