One of the first infinite families of Thue equations, $$F_n(X)=X^3 - (n-1) X^2Y - (n+2)XY^2 - Y^3 = \pm 1$$ for $n\in \mathbb{Z}$, was solved by Thomas in 1990. This family is associated to the simplest cubic fields $\mathbb{Q}(\lambda)$ of Shanks, where $\lambda$ is a root of $F_n(X,1)$. Levesque and Waldschmidt twisted the Thue equations by an exponent $t$ and looked at the equation $$N_{\mathbb{Q}(\lambda)/\mathbb{Q}}(X-\lambda^t Y)=\pm 1,$$ where $t\in \mathbb{Z}$ with $t\neq 0$. In this paper, we find all solutions $(X,Y,n,t)\in \mathbb{Z}^4$ with $n,t\in\mathbb{Z}$ and $t\neq 0$ to this family of twisted Thue equations, thereby answering a question of Levesque and Waldschmidt.
Let $\kk$ be a \emph{field of characteristic $p>0$, not assumed algebraically closed}, and let $V=\kk[x]/(x^p)$ be the Novikov algebra with product $a\circ b=ab'$. For $\lambda\in\kk$, let $M(\lambda)$ be Xu's module. We compute the second cohomology $\Ht(V,M(\lambda))$ for all $\lambda$ and $p$, and describe the assoc...
Let $T(x)=\prod_{j\ge 0}(1-x^{2^j})=\sum_{n\ge 0}t(n)x^n$ be the Thue--Morse generating function, and write $T(x)^m=\sum_{n\ge 0}t_m(n)x^n$ for a positive integer $m$. For a nonzero integer $a$, let $\nu_2(a)$ be the exponent of $2$ in $a$ and put $\operatorname{odd}(a)=a/2^{\nu_2(a)}$. For every $s\ge 1$, we prove tha...
We solve the long-standing problem of determining the exact denominators of the coefficients of the logarithm $B(x)=x+\sum_{n\geq1}b_nx^{n+1}$, obtained from the universal formal group of complex cobordism by the modulus square construction: $$ b_n=\frac{C_n}{d_n},\qquad d_n=\frac{n+1}{2}\operatorname{lcm}(1,\ldots,2n+...
We determine all solutions of
\[
x^2+3^a7^b37^c=\lambda y^n,
\]
where $\lambda\in\{1,2,4\}$, $x,y\geq1$, $a,b,c\geq0$, $n\geq3$, and $\gcd(x,y)=1$, subject to the following parity condition: when $\lambda\in\{1,2\}$ and neither $3$ nor $4$ divides $n$, the integer $y$ is assumed to be odd. The cases divisible by $3$ or...
Mustafa Aydın, M. Alan· Earthline Journal of Mathema...· 0 citations
For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schr\"{o}dinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s(\mathbb{R}^n)$ that guarantees pointwise convergence of $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resol...
Let $p$ and $\ell$ be distinct odd primes. For a finite extension $F/\mathbb{Q}_p$, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of $\text{GL}_n(F)$ and $n$-dimensional, $\Phi$-semisimple Weil--Deligne representations of the Weil group $W_F$....
Imin Chen, Deniz Suozer· 0 citations
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