Mathematicians have been attempting to find better and faster ways to factor composite numbers since the beginning of time. Initially this involved dividing a number by larger and larger primes until you had the factorization. This trial division was not improved upon until Fermat applied the

me. Even in the first grade, my friend Coleman and I used to race each other to see who could finish our pages of addition and subtraction and get the most answers right. (I usually won, but he’s now president of a very large jewelry store. Oh, well....) In third grade, my friend Ron and I decided to write down all the Roman numerals up to 1000, just for the fun of it. And after long division in the fourth grade, I stood atop Arithmetic with no new worlds to conquer. Or so I thought. Right at the end of the fifth grade, we were told to read the Iliad and the Odyssey over the summer, and that in sixth grade math, Mrs. Garrison was going to teach us how to take square roots by hand. “Great! ” I said. But what was a square root? I soon found out. We children who grew up in New Orleans and who rode on truck floats on Mardi Gras Day all knew about perfect squares. A gross was a square, namely 12 times 12. You bought carnival throws and beads by the gross, and every kid knew that a gross was a dozen dozen, or 12 times 12, or 144. We soon learned that a square root was a number, such as 12, that you multiplied by

It is often the case that seemingly unrelated parts of mathematics turn out to have unexpected connections. In this paper, we explore three puzzles and see how they are related to continued fractions, an area of mathematics with a distinguished history within the world of number theory.

Construct a recursive sequence of polynomials, staring with 1, in the following way. Each new term in the sequence is determined by adding the smallest power of x larger than the degree of the previous term, such that the new polynomial is reducible over the rationals. Filaseta, Finch and Nicol have shown that this sequence is finite. In this paper we investigate variations of this problem over a finite field. In particular, we allow the starting polynomial to be any {0, 1}–polynomial with nonzero constant term, and we allow the exponent on the power of x added at each step to be chosen from the set of multiples of a fixed positive integer k. Among our results, we show that these sequences are always infinite. We develop necessary and sufficient conditions on k and the characteristic p of the field, so that the sequence starting with 1 uses every multiple of k as an exponent in its construction. In addition, we prove for k = 1 and p ≥ 5 that there exists a {0, 1}–polynomial f such that the sequence starting with f uses every positive integer larger than the degree of f as an exponent in its construction. 1 1

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These course notes have been developed over several years, using material from many sources, regretfully only some of which are directly acknowledged.

Number theory is an ancient area of mathematics. Carl Friedrich Gauss (1777-1855), whose work defines where elementary number theory turned into modern number theory, is quoted (by G.H. Hardy, in “A Mathematician’s Apology ” (1940)) as saying “Mathematics is the queen of the sciences and number theory is the queen of mathematics.”

Abstract. A conjecture of Neumann and Wahl is confirmed with an explicit example. Some computational techniques are developed, and basic facts about cusps are reviewed. 1.