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Full-Text Articles in Number Theory
A (Not So) Complex Solution To A² + B² = Cⁿ, Arnold M. Adelberg, Arthur T. Benjamin, David I. Rudel '99
A (Not So) Complex Solution To A² + B² = Cⁿ, Arnold M. Adelberg, Arthur T. Benjamin, David I. Rudel '99
All HMC Faculty Publications and Research
No abstract provided in this article.
Nonic 3-Adic Fields, John W. Jones, David P. Roberts
Nonic 3-Adic Fields, John W. Jones, David P. Roberts
Mathematics Publications
We compute all nonic extensions of Q3 and find that there are 795 of them up to isomorphism. We describe how to compute the associated Galois group of such a field, and also the slopes measuring wild ramification. We present summarizing tables and a sample application to number fields.
An Abc Construction Of Number Fields, David P. Roberts
An Abc Construction Of Number Fields, David P. Roberts
Mathematics Publications
We describe a general three step method for constructing number fields with Lie-type Galois groups and discriminants factoring into powers of specified primes. The first step involves extremal solutions of the matrix equation ABC = I. The second step involves extremal polynomial solutions of the equation A(x) + B(x) + C(x) = 0. The third step involves integer solutions of the generalized Fermat equation axp + byq + czr = 0. We concentrate here on details associated to the third step and give examples where the field discriminants have the form ±2a3b .
Random Walks With Badly Approximable Numbers, Doug Hensley, Francis Su
Random Walks With Badly Approximable Numbers, Doug Hensley, Francis Su
All HMC Faculty Publications and Research
Using the discrepancy metric, we analyze the rate of convergence of a random walk on the circle generated by d rotations, and establish sharp rates that show that badly approximable d-tuples in Rd give rise to walks with the fastest convergence.
Pencils Of Quadratic Forms Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas
Pencils Of Quadratic Forms Over Finite Fields, Robert W. Fitzgerald, Joseph L. Yucas
Articles and Preprints
A formula for the number of common zeros of a non-degenerate pencil of quadratic forms is given. This is applied to pencils which count binary strings with an even number of 1's prescribed distances apart.
Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin
Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin
Mathematics Faculty Publications
In this paper we give a new formula for the n-th power of a 2 × 2 matrix. More precisely, we prove the following: Let A = (a b c d) be an arbitrary 2 × 2 matrix, T = a + d its trace, D = ad − bc its determinant and define yn : = b X n/2c i=0 (n − i i )T n−2i (−D) i . Then, for n ≥ 1, A n = (yn − d yn−1 b yn−1 c yn−1 yn − a yn−1) . We use this formula together with an existing formula …