a. w. walker

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Counting Real Quadratic Fields Without Unique Factorization

February 5, 2017by Alexander Walker Leave a comment

The full list of imaginary quadratic fields with unique factorization has been determined, but it remains unknown whether or not infinitely many real quadratic fields are UFDs. Here, we develop a simple criterion for disproving unique factorization based on continued fractions and use this to give upper bounds.

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number theory

The Orders of Simple Groups

February 5, 2017by Alexander Walker 2 Comments

The Classification of Finite Simple Groups proves in particular that the collection of orders of finite simple groups has asymptotic density 0. What can we prove if we’re not willing to work that hard?

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group theory, number theory, probability

Permutations of Maximal Order

February 5, 2017by Alexander Walker 1 Comment

Landau considered the following question – What is the maximal order of an element in the permutation group on k letters? We’ll prove some elementary bounds and deduce an asymptotic using the PNT.

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group theory, number theory

Loaded Dice and Cyclotomic Polynomials

February 5, 2017by Alexander Walker Leave a comment

Does there exist a pair of loaded six-sided dice such that the probability of rolling any dice sum {2,..12} is equally likely? We’ll show how this and other related questions about dice sums can be analyzed using cyclotomic polynomials.

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number theory, probability

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You have stumbled upon the personal blog of Alex Walker. I am a number theorist and a Heilbronn fellow at UCL.

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