In particular, B^\0) = B^\ the Bernoulli number of order k, and BJp = Bn, the ordinary Bernoulli number. Note also that B^ = 0 for n > 0. The polynomials B^\z) and the numbers B^ were first defined and studied by Niels Norlund in the 1920s; later they were the subject of many papers by L. Carlitz and others. For the past twenty-five years not much has been done with them, although recently the ...

Let F q be a finite field of order q with q = p n , where p is a prime. A multiplicative character χ is a homomorphism from the multiplicative group F * q , ·· to the unit circle. In this note we will mostly give a survey of work on bounds for the character sum x χ(x) over a subset of F q. In Section 5 we give a nontrivial estimate of character sums over subspaces of finite fields. §1. Burgess'...

Zaremba conjectured that given any integer m > 1 , there exists an integer a < m with a relatively prime to m such that the simple continued fraction [0, cx.cr] for a/m has c¡ < B for i = 1, 2,..., r, where B is a small absolute constant (say 5 = 5). Zaremba was only able to prove an estimate of the form c, < C log m for an absolute constant C . His first proof only applied to the case where m ...

It is shown that if the first case of Fermat's last theorem fails for an odd prime /, then the sums of reciprocals modulo /, s(k, N) = £ 1/7 (kl/N < j < (k + 1)//A0 are congruent to zero mod/ for all integers N and k with I < N < 46 and 0 < k < N 1 . This is equivalent to Bi_{(k/N) B¡_x =0 (mod/), where B„ and B„(x) are the «th Bernoulli number and polynomial, respectively. The work can be cons...

let g be a (p, q) graph. let f : v (g) → {1, 2, . . . , k} be a map. for each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of g if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

| The well known \strong pseudoprime test" has its highest probability of error (1=4) when the numbers being tested are certain Carmichael numbers with 3 prime factors. We present a non-rigorous plausibility argument that the count C3 (x) of 3-factor Carmichaels up to x, is asymptotically C3(x) K x 1=3 (ln x) ?3 where K 478 110 is an absolute constant given by K = 3 3 X 1a<b<c 1 (abc) 4=3 Y p F...

Solution 1.1.1. The prime factorization of 720 is 2 · 3 · 5. If we look only at the prime factors of x, y, z, we see that the exponents of 2 in the factorizations of x, y, z must sum to 4, the exponents of 3 must sum to 2, and the exponents of 5 must sum to 0. The number of possible triples (a, b, c) that correspond to the exponents of 2 in the prime factorizations of x, y, z is simply the numb...

Abstract We report a search for heavier partner of the recently observed state, denoted as Z_{cs}^{\prime -} , in process e^{+} e^{-}\rightarrow K^{+}D_{s}^{*-}D^{* 0}+c.c. based on e^+e^- collision data collected at center-of-mass energies \sqrt{s}=4.661 4.682 and 4.699 GeV with BESIII detector. The is interest it expected to be candidate hidden-charm open-strang...