Related papers: On a conjecture of Sun

We establish an explicit connection between a Davenport expansion and the Popov sum. Asymptotic analysis follows as a result of these formulas. New solutions to a query of N.J. Fine are offered, and a proof of Davenport expansions is…

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

In this paper we extend methods of Rubin to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field k and prime numbers p which divide the number of roots of unity in k.

For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2},…

We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4. In general, for…

Monin and Rana conjectured a set of equations defining the image of the moduli space $\bar{M}_{0,n}$ under an embedding into $\mathbb{P}^1\times \cdots\times \mathbb{P}^{n-3}$ due to Keel and Tevelev and verified the conjecture for $n\leq…

We present a simple proof of the fact that the base (and independence) polytope of a rank $n$ regular matroid over $m$ elements has an extension complexity $O(mn)$.

In this work, we establish modular parameterizations for two general formulas for $\frac{1}{\pi}$ that subsume conjectural Ramanujan type formulas due to Z.-W. Sun, which have remained open since 2011. As an application of this, in a…

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$…

In this note, we evaluate a series for $1/\pi$ conjectured by Sun. Our proof uses the Cauchy product and hypergeometric transformations. From this result, we derive two additional analogous series for $1/\pi$ involving polynomials of degree…

In this paper, we will give an extension of Mok's theorem on the generalized Frankel conjecture under the condition of the orthogonal bisectional curvature.

We mainly show a supercongruence for a truncated series with cubes of Catalan numbers which extends a result by Zhi-Wei Sun.

In his study of Ramanujan-Sato type series for $1/\pi$, Sun introduced a sequence of polynomials $S_n(q)$ as given by $$S_n(q)=\sum\limits_{k=0}^n{n\choose k}{2k\choose k}{2(n-k)\choose n-k}q^k,$$ and he conjectured that the polynomials…

Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \Q(\sqrt{-nm}) $ and the digits of the…

We consider restricted sumsets over field $F$. Let\begin{align*}C=\{a_1+\cdots+a_n:a_1\in A_1,\ldots,a_n\in A_n, a_i-a_j\notin S_{ij}\ \text{if}\ i\not=j\},\end{align*} where $S_{ij}(1\leqslant i\not=j\leqslant n)$ are finite subsets of $F$…

We prove Ma\~n\'e's conjectures in the context of codimension one Aubry-Mather theory

In this paper, we shall prove two conjectures of Z.-W. Sun concerning Ap\'ery-like series. One of the series is alternating whereas the other one is not. Our main strategy is to convert the series (resp.~the alternating series) to…

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then we have $$…

We present a conjecture about partitions, with a very elementary formulation.

The Franel numbers are defined by $ f_n=\sum_{k=0}^n {n\choose k}^3. $ Motivated by the recent work of Z.-W. Sun on Franel numbers, we prove that \begin{align*} \sum_{k=0}^{n-1}(3k+1)(-16)^{n-k-1} {2k\choose k} f_k &\equiv…