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 Fq, 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!
II. Math Book Errata
- Koblitz, A Course in Number Theory and Cryptography, 2nd ed. (GTM 114)
[ERRATA]
© 2002-2008 H. Timothy Choi