The coefficients c(n, k) defined by (1− kx) = X n≥0 c(n, k)x reduce to the central binomial coefficients `

We explore a conjecture posed by Eswarathasan and Levine on the distribution of p-adic valuations harmonic numbers $$H(n)=1+1/2+\cdots +1/n$$ that states set $$J_p$$ positive integers n such p divides numerator H(n) is finite. proved two results, using modular-arithmetic approach, one for non-Wolstenholme primes other Wolstenholme primes, an anomalous asymptotic behaviour valuation $$H(p^mn)$$ ...

We give a proposal for future development of the model theory of valued fields. We also summarize recent results on p-adic numbers. Let K be a valued field with a valuation map v : K → G ∪ {∞} to an ordered group 1 G; this is a map satisfying (i) v(x) = ∞ if and only if x = 0; (ii) v(xy) = v(x) + v(y) for all x, y ∈ K; (iii) v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K. We write R for the valuat...

The idea is simple: we want to develop a theory of analytic manifolds and spaces over fields equipped with an arbitrary complete valuation. Of course, it is a standard fact that such a field must be either R, C, or a field with a nonarchimedean valuation, so what we really mean is that we want to develop a theory of nonarchimedean analytic spaces. Doing this näıvely (i.e., defining manifolds in...

from which it follows that dl(m) is a rational number with only a power of 2 in its denominator. Extensive calculations have shown that, with rare exceptions, the numerators of dl(m) contain a single large prime divisor and its remaining factors are very small. For example d6(30) = 2 12 · 7 · 11 · 13 · 17 · 31 · 37 · 639324594880985776531. Similarly, d10(200) has 197 digits with a prime factor ...

Let V (resp. D) be a valuation domain (resp. SFT Prüfer domain), I a proper ideal, and V̂ (resp. D̂) be the I-adic completion of V (resp. D). We show that (1) V̂ is a valuation domain, (2) Krull dimension of V̂ = dimV I+1 if I is not idempotent, V̂ ∼= V I if I is idempotent, (3) dim D̂ = dimD I + 1, (4) D̂ is an SFT Prüfer ring, and (5) D̂ is a catenarian ring. Throughout this paper, all rings are assu...

Abstract. Due to its fractal nature, much about the area of the Mandelbrot set M remains to be understood. While a series formula has been derived by Ewing and Schober to calculate the area of M by considering its complement inside the Riemann sphere, to date the exact value of this area remains unknown. This paper presents new improved upper bounds for the area based on a parallel computing al...

The integral representation algebra A(RG) is C ®z a(RG). When does a(RG) contain nontrivial nilpotent elements? Let | G\ = pn, where p\n, p prime. Denote by Zp the £-adic valuation ring in Q, and by Zp* its completion. Reiner has shown (i) If a = l , then A(ZPG) and A(Z*G) have no nonzero nilpotent elements (see [ l ] ) . (ii) If ce^2, and G has an element of order p, then both A(ZPG) and A{Z*G...

In this paper, we investigate the 2-adic valuation of the Stirling numbers S(n, k) of the second kind. We show that v2(S(2n + 1, k + 1)) = s2(n) − 1 for any positive integer n, where s2(n) is the sum of binary digits of n. This confirms a conjecture of Amdeberhan, Manna and Moll. We show also that v2(S(4i, 5)) = v2(S(4i + 3, 5)) if and only if i 6≡ 7 (mod 32). This proves another conjecture of ...

Let (K, v) be a discrete valued field with valuation ring O, and let Ov be the completion of O with respect to the v-adic topology. In this paper we discuss the advantages of manipulating polynomials in Ov [x] on a computer by means of OM representations of prime (monic and irreducible) polynomials. An OM representation supports discrete data characterizing the Okutsu equivalence class of the p...