arXiv subjects
Publications and source records attributed to A. L. P. Porto.
We study the profinite genus of HNN-extensions whose associated subgroups are finite. We give precise formulas for the number of isomorphism classes of HNN(G,H,K,t,f) and of its profinite completion and compute the profinite genus of such an HNN-extension HNN(G,H,K,t,f). We also list various situations when HNN(G,H,K,t,f) is determined by its profinite completion.
In this paper we find non-negative integer solutions for exponential Diophantine equations of the type $p \cdot 3^x+ p^y=z^2,$ where $p$ is a prime number. We prove that such equation has a unique solution $\displaystyle{(x,y,z)=\left(\log_3(p-2), 0, p-1\right)}$ if $2 \neq p \equiv 2 \pmod 3$ and $(x,y,z)=(0,1,2)$ if $p=2$. We also display the infinite solution set of that equation in the case $p=3$. Finally, a brief discussion of the case $p \equiv 1 \pmod 3$ is made, where we display an equation that does not have a non-negative integer solution and leave some open questions. The proofs are based on the use of the properties of the modular arithmetic.