Related papers: On two congruences involving Franel numbers

We prove that the Farey sequences can be express into equivalence classes labeled by a fractal parameter which looks like a Hausdorff dimension $h$ defined within the interval 1 < h < 2. The classes $h$ satisfy the same properties of the…

Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in…

In this paper we shall be concerned with $H_\alpha$ summability, for $0 < \alpha \leq 2$ of the Fourier series of arbitrary $L^1([-\pi,\pi]) $ functions. The methods to be employed here are a refinement of the real variable methods…

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n-$th Bernoulli numbers. In this paper, we will generalize…

In this paper, we employ the theories and techniques of hypergeometric functions to provide two distinct proofs of the conjectured identities involving multiple Ap\'ery-like series with central binomial coefficients and multiple harmonic…

We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of…

We present a new formula for pi involving nested radicals with rapid convergence. This formula is based on the arctangent function identity with argument $x=\sqrt{2-{{a}_{k-1}}}/{{a}_{k}}$, where \[…

In this paper, we prove a conjecture by Daniele Mundici on the sum of squared distances between consecutive elements in the $Q$-th Farey sequence for $Q\in\mathbb{Z}$ and $Q\geq 2$.

Recently the author proved that the 1977 Hummel-Scheinberg-Zalcman conjecture on coefficients of nonvanishing $H^p$ functions is true for all $p = 2m, m \in \mathbb{N}$, i.e., for the Hilbertian Hardy spaces $H^{2m}$. As a consequence, this…

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by…

In this paper, we mainly prove the following congruence conjectured by J.-C. Liu: $$ {}_6F_5\bigg[\begin{matrix}\frac{5}{4}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\&\frac{1}{4}&1&1&1&1\end{matrix}\bigg|\…

We offer a mathematical proof of consistency for Peano Arithmetic PA formalizable in PA. This result is compatible with Goedel's Second Incompleteness Theorem since our consistency proof does not rely on the representation of consistency as…

We use the results of our paper "p-Fractals and power series--I" (Journal of Algebra 280, 2004, pp. 505--536) to prove the rationality of the Hilbert-Kunz series of a large family of power series, including those of the form \sum_i…

The generalized summation of divergent trigonometric series, namely by method of $\sigma_k(r,a)$-factors is considered in this paper. It is proved that such summation of Fourier series of periodical function $f(t)$ results in the…

We study the arctanh sums h(k) = sum_{n=2}^\infty arctanh(n^{-k}) as a function of a complex variable k. Building on the closed-form identity h(k) = (1/2) log(g(2k)/g(k)^2) (proved in the companion preprint arXiv:2602.06244), we develop the…

Let p be any odd prime. We mainly show that $$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and $$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order…

Let $A$ be a discrete valuation ring with field of fractions $F$ and (sufficiently large) residue field $k$. We prove that there is a natural exact sequence $H_3(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}]) \to…

We prove Hunter's conjecture on complete homogeneous symmetric polynomials. For even $n$ and every integer $k\geq 1$, we show that under the constraint $\sum_{i=1}^n a_i^2=1$ the global minimum of the even-degree polynomial…

We prove hypergeometric type summation identities for a function defined in terms of quotients of the $p$-adic gamma function by counting points on certain families of hyperelliptic curves over $\mathbb{F}_{q}$. We also find certain special…

Let $k\ge 2$ and $\{F_n^{(k)}\}_{n\geq 2-k}$ be the sequence of $k$--generalized Fibonacci numbers whose first $k$ terms are $0,\ldots,0,0,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we determine all…