arXiv subjects
Publications and source records attributed to Ömer Avcı.
If $a>b$ and $n>1$ are positive integers and $a$ and $b$ are relatively prime integers, then a large Zsigmondy prime for $(a,b,n)$ is a prime $p$ such that $p \,|\, a^n-b^n$ but $p \,\nmid \, a^m-b^m$ for $1 \leq m < n$ and either $p^2 \, | \, a^n - b^n$ or $ p > n + 1$. We classify all the triples of integers $(a, b, n)$ for which no large Zsigmondy prime exists.
Apollonius of Perga, showed that for two given points $A,B$ in the Euclidean plane and a positive real number $k\neq 1$, geometric locus of the points $X$ that satisfies the equation $|XA|=k|XB|$ is a circle. This circle is called Apollonius circle. In this paper we generalize the definition of the Apollonius circle for two given circles $Γ_1,Γ_2$ and we show that geometric locus of the points $X$ with the ratio of the power with respect to the circles $Γ_1,Γ_2$ is constant, is also a circle. Using this we generalize the definition of Apollonius Circle, and generalize some results about Apollonius Circle.