ترغب بنشر مسار تعليمي؟ اضغط هنا

Variants of Base 3 over 2

62   0   0.0 ( 0 )
 نشر من قبل Tanya Khovanova
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

We discuss two different systems of number representations that both can be called base 3/2. We explain how they are connected. Unlike classical fractional extension, these two systems provide a finite representation for integers. We also discuss a connection between these systems and 3-free sequences.



قيم البحث

اقرأ أيضاً

We discuss properties of integers in base 3/2. We also introduce many new sequences related to base 3/2. Some sequences discuss patterns related to integers in base 3/2. Other sequence are analogues of famous base-10 sequences: we discuss powers of 3 and 2, Look-and-say, and sorted and reverse sorted Fibonaccis. The eventual behavior of sorted and reverse sorted Fibs leads to special Pinocchio and Oihcconip sequences respectively.
We delve into the connection between base $frac{3}{2}$ and the greedy partition of non-negative integers into 3-free sequences. Specifically, we find a fractal structure on strings written with digits 0, 1, and 2. We use this structure to prove that the even non-negative integers written in base $frac{3}{2}$ and then interpreted in base 3 form the Stanley cross-sequence, where the Stanley cross-sequence comprises the first terms of the infinitely many sequences that are formed by the greedy partition of non-negative integers into 3-free sequences.
We give a characterization of all matrices $A,B,C in mathbb{F}_{2}^{m times m}$ which generate a $(0,m,3)$-net in base $2$ and a characterization of all matrices $B,Cinmathbb{F}_{2}^{mathbb{N}timesmathbb{N}}$ which generate a $(0,2)$-sequence in base $2$.
Let $pequiv 1,(mathrm{mod},9)$ be a prime number and $zeta_3$ be a primitive cube root of unity. Then $mathrm{k}=mathbb{Q}(sqrt[3]{p},zeta_3)$ is a pure metacyclic field with group $mathrm{Gal}(mathrm{k}/mathbb{Q})simeq S_3$. In the case that $mathrm {k}$ possesses a $3$-class group $C_{mathrm{k},3}$ of type $(9,3)$, the capitulation of $3$-ideal classes of $mathrm{k}$ in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-$3$-extension $mathrm{k}_3^{(infty)}$ of $mathrm{k}$ are drawn.
In the spirit of Lehmers unresolved speculation on the nonvanishing of Ramanujans tau-function, it is natural to ask whether a fixed integer is a value of $tau(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We offer a method, w hich applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3leq ellleq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $tau(n)$ with $n>1$, we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 13, pm 17, -19, pm 23, pm 37, pm 691},$$ and assuming GRH we show for primes $ell$ that $$tau(n) ot in left { pm ell : 41leq ellleq 97 {textrm{with}} left(frac{ell}{5}right)=-1right} cup left { -11, -29, -31, -41, -59, -61, -71, -79, -89right}. $$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $ell$, we prove that $pm ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{pm}(ell,m)=O_{ell}(m)$ and even level coprime to $ell,$ where $M^{pm}(ell,m)$ is effectively computable.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا