The discussion centers on proving that all points inside a circle share the same center using the $p$-adic metric defined as $d_p(m,n)=|n-m|_p$. Participants clarify the definition of a circle's center and correct the metric notation. The $p$-adic metric is specifically applied to the rational numbers, $\mathbb{Q}$, and a reference is provided for further reading on the topic.
PREREQUISITESMathematicians, students studying number theory, and anyone interested in the properties of $p$-adic metrics and their applications in geometry.
Could you give the definition of a circle center? Also, a metric is a function of two arguments, while $d_p(n+m)$ has one argument. Finally, what is $p$?Julio said:Show that all point inside of an circle is his center. Consider the metric $d_p(n+m)=|n-m|_p.$
I suppose you mean $d_p(m,n)=|n-m|_p$ where $d_p$ is the $p$-adic metric on $\mathbb{Q},$ and disc instead of circle. If so, have a look https://www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf.Julio said:Show that all point inside of an circle is his center. Consider the metric $d_p(n+m)=|n-m|_p.$