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Abstract:
In 1980, Pomerance and Selfridge proved D. J. Newman's coprime
mapping conjecture: If N is a positive integer and I is a set of
N consecutive integers, then there is a bijection f:
{1,2,…,N}→ I such that gcd(i, f(i))=1 for 1≤
i≤
N. The function f described in their theorem is called a coprime
mapping. Around the same time, Entringer conjectured that all trees are
prime, that is, the vertex set V can be labeled with the distinct
integers 1, 2,…, |V| such that for each edge xy the labels
assigned to vertices x and y are coprime. So far, there has been
little progress towards a proof of this conjecture. In this talk, I will
prove a generalization of Newman's conjecture and use the result to show
that various families of trees are prime (including regular firecrackers,
banana trees, binomial trees, and families of spider colonies, all of
which will be defined). This is joint work with Ben Small, a senior
mathematics major at Seattle University.
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