Let $(a,b,c,d) = \left(2mn, 2mp, m^2 - n^2 - p^2, m^2 + n^2 + p^2\right)$ be a primitive Pythagorean quadruple and let $S=\langle a, b, c, d\rangle$ be the numerical semigroup generated by $a,b,c,$ and $d.$ For convenience, we also let $Q = n^2 + p^2, \delta = \gcd(n,p),$ and $n = \delta n_0$ for some $n_0 \in \mathbb{Z}$. In this paper, we determine the Frobenius number of $S$ and derive an explicit formula in terms of $m$, $n$, and $p$ with the assumption that $m\geq 2Q$ for $\delta = 1$ and $m \geq \frac{2Q}{\delta} - 1$ for the remaining cases. The proof is based on an explicit complete residue system modulo $2mn$, a normalization procedure for arbitrary semigroup elements, and a lift-orbit description of boundary representatives.
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
Let $k$ be a field of characteristic $p\ge0$. We prove that $m$ points and $n$ lines in $k^2$ determine at most $3(mn)^{2/3}+m+n+2mn/p$ incidences, the last term being omitted in characteristic zero. Over the prime field $\mathbb{F}_p$ the coefficient of $mn/p$ can be replaced by $1$. The proof uses the polynomial meth...
Let $b, p\in {\mathbb Z}$ with $b\ge 2$ and $p\ge 3$ a prime. If $b^{p-1}\equiv 1 \pmod{p^2}$, then $p$ is called a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime}. When $b=2$, $p$ is also known simply as a Wieferich prime. We say that a monic polynomial $f(x)\in {\mathb...
Let $F$ be a field of characteristic zero and consider $M_2(F),$ the algebra of $2\times 2$ matrices over $F,$ with canonical $\mathbb{Z}_2$-grading and endowed with transpose superinvolution. In this paper, we present the generators of the $T_2^*$-ideal of $*$-identities and of the $T_2^*$-subspace of central $*$-poly...
Let $S$ be a free profinite group on a finite ordered basis $X$, and let $S^{(n,p)}$, $n=1,2,\ldots,$ denote its lower $p$-central filtration. There is a natural duality between $S^{(n,p)}/S^{(n+1,p)}$ and $H^2(S/S^{(n,p)},\mathbb{F}_p)$. These $\mathbb{F}_p$-linear spaces admit natural bases indexed by Lyndon words of...
Let $c\phi_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\geq2$ and every $k\equiv2\pmod{2^m}$, \[ \sum_{n\geq0}c\phi_k(n)q^n\equiv\frac{\varphi(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}c\phi_{k/2}(n)q^{2n}\pmod{2^m}, \] where $\varphi(q)$ is the cla...
Manjil P. Saikia· 0 citations
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