A proof of Hadjicostas's conjecture

Atkin-O'Brien 分拆同余猜想关于13的幂次的一个简单证明

A simple proof of the Atkin-O'Brien partition congruence conjecture for powers of 13

本文针对1967年Atkin和O'Brien提出的涉及Hecke算子模13的幂次的分拆函数同余猜想，给出了一个简单证明。

About the Proof of Lang's Height Conjecture

This is an informal paper presenting historical results around the recent paper of the author about Lang's Conjecture and torsion of elliptic curves. This paper also discusses a few aspects of the proof.

Proof of the conjecture of Keskin, Siar and Karaatli

In this paper among other results, we will prove the conjecture of Keskin, Siar and Karaatli on the Diophantine equation $x^{2}-kxy+y^{2}-2^{n}=0$.

Proof of conjecture regarding the level of Rose's generalized sum-of-divisor functions

In a recent paper, Rose proves that certain generalized sum-of-divisor functions are quasi-modular forms for some congruence subgroup and conjectures that these forms are quasi-modular for $Γ_1(n)$. Here, we prove this conjecture.

An algorithmic proof of Bachet's conjecture and the Lagrange-Euler method

The goal of this notice is to present a proof of Bachet's conjecture based exclusively on the fundamental theorem of arithmetic. The novelty of this proof consists in its introduction of a partial order on rational integers through the unique factorization property. In general, the proofs of Bachet's conjecture by Lagrange - Euler's method assume necessary the use of infinite descent. In the proposed proof we do not assume the existence of a "minimal solution", but rather we show the existence of the desired solution through an algorithmic method.

Proof of Cluckers-Veys's conjecture on exponential sums for polynomials with log-canonical threshold at most a half

In this paper, we will give two proofs of the Cluckers-Veys conjecture on exponential sums for the case of polynomials in $\mathbb{Z}[x_{1},\ldots,x_{n}]$ having log-canonical thresholds at most one half. In particular, these results imply Igusa's conjecture and Denef-Sperber's conjecture under the same restriction on the log-canonical threshold.

广义过立方划分结果的组合证明及相关猜想

Combinatorial proof of a result on generalized overcubic partitions and related conjectures

本文通过组合方法证明了广义过立方划分在模4下的结果，并验证了相关猜想。

Algorithmic Proof of the Epsilon Constant Conjecture

In this paper we will algorithmically prove the local and global epsilon constant conjectures for all fields of absolute degree lower or equal to 15. To this end we will present an efficient algorithm for the computation of local fundamental classes and address several other problems arising in the algorithmic proof.

有限域上本原四次多项式的一个猜想的证明

Proof of a Conjecture on Primitive Quartic Polynomials over Finite Fields

针对有限域上本原四次多项式的猜想，本文在q足够大时，借助Fu和Wan的定理，通过AI辅助完成了证明。

Appendix to: The level 1 weight 2 case of Serre's conjecture - a strategy for a proof

In this appendix, we observe that our March preprint on Serre's conjecture was indeed correct: the only "missing argument" follows automatically from a result of Bockle and Ramakrishna. Thus, we get a proof of the level 1 weight 2 case of Serre's conjecture (a result that has been also proved independently by Khare and Wintenberger).

A Proof of Merca's Conjectures on Sums of Odd Divisor Functions

In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these conjectures.

A remark on a recent proof of Lehmer's Conjecture [arXiv:1709.03771]

Over one year ago, a very long preprint posted on arXiv [arXiv:1709.03771] and HAL announced a proof of Lehmer's Conjecture (and of other related results). Unfortunately, as was remarked by several specialists, this proof contains a (at least one) fatal error. The aim of this very short note is to inform the mathematical community, which could be aware of this.

A proof of the Howe duality conjecture

We give a proof of the Howe duality conjecture in local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic.

Proofs of some partition identities conjectured by Kanade and Russell

Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey-Wilson and Rogers polynomials. We also obtain some related results, including a new proof of a partition identity conjectured by Capparelli and fir…

论沃伊塔对莫德尔猜想的证明

On Vojta's proof of the Mordell conjecture

本文用阿拉克洛夫几何重构沃伊塔对莫德尔猜想的证明，并以袁的算术萧不等式替代吉莱--苏莱算术黎曼--罗赫定理。

On Hrushovski's proof of the Manin-Mumford conjecture

The Manin-Mumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening result using classical algebraic geometry alone. Altogether, this yields a new proof of the Manin-Mumford conjecture using only classical algebraic geometry.

A proof of Ryser's circulant Hadamard conjecture

We show that an $n\times n$ circulant Hadamard matrix must satisfy a family of congruence equations that have solutions only when $n \leq 4$, proving Ryser's 1963 conjecture that no such matrices exist for $n>4$.

A proof of two conjectures on APN functions

Dobbertin, Mills, Müller, Pott and Willems conjecture that two families of power mapping are families of APN functions. Here we prove those two conjectures.

Remarks to Arsovski's proof of Snevily's conjecture

Based on the recent work of Arsovski, we confirm a conjecture of Feng, Sun, and Xiang, and we give a shortened proof of Snevily's conjecture.