ترغب بنشر مسار تعليمي؟ اضغط هنا

On Generalized Carmichael Numbers

97   0   0.0 ( 0 )
 نشر من قبل Tae Kyu Kim
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English
 تأليف Yongyi Chen




اسأل ChatGPT حول البحث

Given an integer $k$, define $C_k$ as the set of integers $n > max(k,0)$ such that $a^{n-k+1} equiv a pmod{n}$ holds for all integers $a$. We establish various multiplicative properties of the elements in $C_k$ and give a sufficient condition for the infinitude of $C_k$. Moreover, we prove that there are finitely many elements in $C_k$ with one and two prime factors if and only if $k>0$ and $k$ is prime. In addition, if all but two prime factors of $n in C_k$ are fixed, then there are finitely many elements in $C_k$, excluding certain infinite families of $n$. We also give conjectures about the growth rate of $C_k$ with numerical evidence. We explore a similar question when both $a$ and $k$ are fixed and prove that for fixed integers $a geq 2$ and $k$, there are infinitely many integers $n$ such that $a^{n-k} equiv 1 pmod{n}$ if and only if $(k,a) eq (0,2)$ by building off the work of Kiss and Phong. Finally, we discuss the multiplicative properties of positive integers $n$ such that Carmichael function $lambda(n)$ divides $n-k$.



قيم البحث

اقرأ أيضاً

78 - Rusen Li 2021
In this paper, we introduce a new type of generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$, and show that Euler sums of the generalized alternating hyperharmonic numbers $H_n^{(p,r,s_{1},s_{2})}$ can be expressed in terms of li near combinations of classical (alternating) Euler sums.
266 - Tomohiro Yamada 2020
Some new results concerning the equation $sigma(N)=aM, sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
172 - Taekyun Kim 2008
Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers.
118 - Taekyun Kim 2008
In this paper we investigate the properties of the Euler functions. By using the Fourier transform for the Euler function, we derive the interesting formula related to the infinite series. Finally we give some interesting identities between the Euler numbers and the second kind stirling numbers.
Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect powers, henc e X is the set {2,3, 5,6,7,10,11,...} . Let M be a function from X to sets of positive integers such that, for each s in X, if m is in M(s) then each divisor of m is in M(s) and if M(s) is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases s^m such that s is in X and m is in M(s). We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.
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