H. Timothy Choi on math.uci.edu server

PROJECT

I. Interesting and Nontrivial (to me) Math Questions

Not that I put significant amount of time but just some random math problems that I usually thought of while driving.

Paper Folding Counting: Take a piece of a rectangular paper. Fold into a half. Then fold into a half. Keep repeating until it becomes too hard to fold into a half. Now unfold until no folding exist. Now it creates a natural rectangular grid on the paper. Besides the way described, there are quite a lot of ways to fold the paper to obtain the smallest rectangular shape. Count the ways.

Normed Space from Metric Space: Given a metric space X with a metric function d, can we construct a normed space V with a norm n such that X is isometrically isomorphic to the metric space V with the metric induced by n.

Additive vs. Multiplicative: Given two field elements a and b in the finite field F_q, what can we say about tr(ab) and/or N(a+b), where tr and N are absolute trace and norm?

Product of Primes + 1: Let p(n) be the product of first n prime numbers. When is p(n) + 1 a prime number? How often p(n) + 1 becomes a prime number? For example, between 1 and 500 there are only 11 values of n such that p(n) + 1 is a prime number. Namely for n = 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, and 457.

p and Sum of Factorials: Show that p does not divide 1! + 2! + … + (p-1)!. It is essentially solved by Bernd C. Kellner from http://www.bernoulli.org/~bk/remkurepa.pdf.

Fractal Primes: 999983 is the largest six-digit prime number (its reversal 389999 is also prime). 999983 is self-similar prime in a sense that the sum of its digits 9 + 9 + 9 + 9 + 8 + 3 = 47 is again a prime, and 4 + 7 = 11 is also prime. Finally 1 + 1 = 2 is prime. Thus, 389999 is also self-similar. Can we find more (or all)?
Blue Region: Find the area of the blue region. The figure is drawn to scale. I got it! As a hint consider the area of the large circle and the sum of small circles.
Is 113 prime?: 13 is prime and 113 is also prime. But 1113 is not. 1113 = (3)(7)(53). When is the integer which is formed by attaching the digits 1 many times in front of 3 prime? The next ones, that are known to me, are 11113 (4 ones), 111111113 (8 ones), 11111111113 (10 ones), 111111111111111111111113 (23 ones), 83 ones, 220 ones, and 1313 ones. Are there infinitely many primes of this form?

Centroid: Consider a scalene triangle ΔABC. Let BD be the median and extend it to BE such that BD = DE. Similarly, let CF be the median and extend it to CG such that CF = FG. Show that GA = EA (without using triangle similarity) and GA and AE are parallel to BC. I got it! For those who are interested in, consider two triangles in color as a hint.

Incenter: Given a right triangle ΔABC with the right angle ∠ABC, let AD, BE, and CF be angle bisectors of the triangle. Their point of concurrency G, called the incenter of the triangle, is equidistant from all three sides of the triangle. Hence, three dashed line segments, altitudes of three triangles ΔAGB, ΔBGC, and ΔCGA, are all congruent. Let x be the length of one of these dashed line segments. Find an equation which involves x, AB, BC, and CA. I got it! As a hint consider the area of the triangles.

II. Math Book Errata

Koblitz, A Course in Number Theory and Cryptography, 2nd ed. (GTM 114) [ERRATA]

III. Some Notes on Mathematics

Wolstenholm's Theorem: The notes has a proof for the theorem and a conjecture based on some calculations.