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Articles 31 - 44 of 44
Full-Text Articles in Number Theory
A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali
A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su
Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su
All HMC Faculty Publications and Research
We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.
Prove It!, Kenny W. Moran
Prove It!, Kenny W. Moran
Journal of Humanistic Mathematics
A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.
Ramanujan Sums As Supercharacters, Christopher F. Fowler '12, Stephan Ramon Garcia, Gizem Karaali
Ramanujan Sums As Supercharacters, Christopher F. Fowler '12, Stephan Ramon Garcia, Gizem Karaali
Pomona Faculty Publications and Research
The theory of supercharacters, recently developed by Diaconis-Isaacs and Andre, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.
The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn
The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn
All HMC Faculty Publications and Research
We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN where N≥ 2. Let ZK be a union of varieties defined over K such that V ⊈ ZK. We prove the existence of a point of small height in V \ ZK, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of hypersurface containing ZK, where dependence on …
Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger
Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger
CMC Senior Theses
Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.
Compression Theorems For Periodic Tilings And Consequences, Arthur T. Benjamin, Alex K. Eustis '06, Mark A. Shattuck
Compression Theorems For Periodic Tilings And Consequences, Arthur T. Benjamin, Alex K. Eustis '06, Mark A. Shattuck
All HMC Faculty Publications and Research
We consider a weighted square-and-domino tiling model obtained by assigning real number weights to the cells and boundaries of an n-board. An important special case apparently arises when these weights form periodic sequences. When the weights of an nm-tiling form sequences having period m, it is shown that such a tiling may be regarded as a meta-tiling of length n whose weights have period 1 except for the first cell (i.e., are constant). We term such a contraction of the period in going from the longer to the shorter tiling as "period compression". It turns out that …
The Probability Of Relatively Prime Polynomials, Arthur T. Benjamin, Curtis D. Bennet
The Probability Of Relatively Prime Polynomials, Arthur T. Benjamin, Curtis D. Bennet
All HMC Faculty Publications and Research
No abstract provided in this article.
Self-Avoiding Walks And Fibonacci Numbers, Arthur T. Benjamin
Self-Avoiding Walks And Fibonacci Numbers, Arthur T. Benjamin
All HMC Faculty Publications and Research
By combinatorial arguments, we prove that the number of self-avoiding walks on the strip {0, 1} × Z is 8Fn − 4 when n is odd and is 8Fn − n when n is even. Also, when backwards moves are prohibited, we derive simple expressions for the number of length n self-avoiding walks on {0, 1} × Z, Z × Z, the triangular lattice, and the cubic lattice.
Some Effective Diophantine Results Over Q-Bar, Lenny Fukshansky
Some Effective Diophantine Results Over Q-Bar, Lenny Fukshansky
CMC Faculty Publications and Research
In his 1999 paper D. W. Masser talks about effective search bounds for polynomial equations over integers and rationals. This discussion can also be extended over number fields. Unfortunately, as illustrated by Matiasevich's negative answer to Hilbert's 10-th problem, search bounds in general probably do not exist. Some special cases are understood, but in general very little is known. I will talk about effective search bounds for solutions of polynomial equations over Q-bar with some additional arithmetic conditions. This discussion also naturaly ties into the realm of "absolute" diophantine results, like Siegel's lemma of Roy and Thunder. I will try …
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.
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.
Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su
Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su
All HMC Faculty Publications and Research
Fix . Consider the random walk on the circle which proceeds by repeatedly rotating points forward or backward, with probability , by an angle . This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy'' distance. The rate depends on the continued fraction properties of the number . We obtain bounds for rates when is any irrational, and a sharp rate when is a quadratic irrational. In that case the discrepancy falls as (up to constant factors), where is the number of steps in the walk. This is the first example of a sharp …