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!
- 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?
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.
© 2002-2008 H. Timothy Choi