Related papers: A proof of the Graham Sloane conjecture
We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre's conjecture follows.
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
Robin's Conjecture is strengthened, deformed, and proved. Nicolas conjecture follows.
We prove the positivity conjecture for all skew-symmetric cluster algebras.
We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of…
A classical probabilistic explanation for Hardy's quantum paradox is demonstrated.
An equivalent but useful version on the Homological Nerve Theorem is proved.
We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these…
We prove the decidability of the elementary theory of a free group.
We give a counterexample to a recently conjectured variant of the Penrose inequality.
We present a new, elementary, dynamical proof of the prime number theorem.
In this article, we prove a weighted version of Saitoh's conjecture. As an application, we prove a weighted version of Saitoh's conjecture for higher derivatives.
We establish an analogue of the Goldbach conjecture for Laurent polynomials with positive integer coefficients.
We show the existence of a set $A\subseteq \mathbb{Z}_{\geq 2}$ satisfying the estimates of the Bateman--Horn conjecture, Goldbach's conjecture, and also \[ \#\{p\leq x \text{ prime} ~|~ p\in A\} \gg x(\log\log x)/(\log x)^2. \]
We prove a local version of the Mazur-Ulam theorem.
We confirm a conjecture of Sun on the expansions of $n(m^k-1)/(m-1)$ in base $m$.
We present a new proof of the celebrated quadratic reciprocity law. Our proof is based on group theory.
In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.
We prove that if the Morrison cone conjecture holds for a smooth Calabi-Yau threefold $Y$, it holds for any smooth Calabi-Yau threefold deformation-equivalent to $Y$. We use this result to prove a new case of the Morrison cone conjecture.
We survey the state of the union-closed sets conjecture.