Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman's coprime mapping conjecture: {\em If $\$n\$$ is a positive integer and $\$I\$$ is a set of $\$n\$$ consecutive integers, then there is a bijection $\$f:\; \backslash \{1,2,\backslash dots,n\backslash \}\backslash rightarrow\; I\$$ such that $\$\backslash gcd(i,\; f(i))=1\$$ for $\$1\backslash leq\; i\backslash leq\; n\$$}. Around the same time, Roger Entringer conjectured that all trees are {\em prime}, that is, that if $\$T\$$ is a tree with vertex set $\$V\$$, then there is a bijection $\$L:\; V\backslash rightarrow\; \backslash \{1,\; 2,\; \backslash ldots,\; |V|\backslash \}\$$ such that $\$\backslash gcd(L(x),\; L(y))=1\$$ for all adjacent vertices $\$x\$$ and $\$y\$$ in $\$V\$$. There has been little progress so far towards a proof of this conjecture. In this talk, I will discuss extensions of Newman's conjecture and how they can be used to prove that various families of trees are prime, including palm trees, banana trees, binomial trees, and certain families of spider colonies.