A couple elementary examples of this: Greatest Common Divisor. Consider the following two statements: S1: The greatest common divisor $d=\gcd(a,b)$ satisfies: (i) $d \mid a$ and $d\mid b$ and (ii) if $...

For any sequence $p_i$ with $\Delta^2 p_i$ the second differences, then $$\sum_{i=1}^n \Delta^2 p_i = p_1-p_2-p_{n+1}+p_{n+2}.$$ So for $p_i$ being the $i$th prime, this becomes $p_{n+2}-p_{n+1}-1.$ ...

This looks similar to numerators of fractions with odd denominator that create specific Collatz cycles. See: Collatz Conjecture: Iterating with odd denominators or 2-adic integers (Wikipedia)

Yes, such a subsequence exists. [Edit: An infinite number of] The continued fraction convergents, $\frac{p_n}{q_n}$, for $\pi/2$ satisfy $$\left|\frac{\pi}{2}-\frac{p_n}{q_n}\right|<\frac{1}{2q_n^2}...

Mathematica (or rather Wolfram Alpha) gives an answer in terms of the digamma function: http://www.wolframalpha.com/input/?i=Sum[1%2F%28a+x%2Bb%29%2C{x%2Cx0%2Cx1}]

Messer and Straffin's book "Topology Now!" provides most of the steps starting from their definition of a knot (able to decomposed into a finite number of linear segments), building up the ideas of ...

Mathematica confirms the following: Change the integral to polar coordinates to get $$\frac{(1)}{2} = \int_0^{\pi/4} \int_0^{\sec(\theta)/2} \frac{1}{1-r^2}r\ dr\ d\theta = \frac{G}{6}.$$

Some of the big ideas and connections (with lots of pertinent references) are presented excellently in Fernando Q. Gouvea's "A Marvelous Proof"

DeMoivre's theorem. The pre-calc version of the proof relies on a lot of triangle geometry to establish trigonometric sum formulas and then uses induction. If you use Euler's Identity, it's a one line ...

See Example B on page 9 of these notes: http://www.math.polytechnique.fr/~chenevier/coursIHP/chenevier_lecture6.pdf Let $D =\left(\frac{−1,−11}{\mathbb{Q}}\right)$ be the quaternion algebra with ...

MacKay and Mahajan have a short easy-to-read article expanding on Noam's comment. (pdf link) Edit (5/14/13): There is a whole bunch of random notes on relationships like this one and others that can ...

The paper "The Natural Chain of Binary Arithmetic Operations and Generalized Derivatives" by M. Carroll (link) is a great paper for undergrads that demonstrates an infinite number of binary operations ...

I've found that doing low-dimensional manifold topology is very appealing to undergraduates. I used the "Topology Now!" text by Messer and Straffin and, while the text isn't perfect, the approach was ...

After a quick glance at your website, I think you should definitely talk about symmetry and invariants and any tie-in to cryptography. The general audience loves visual symmetry and you may actually ...

Your proto-Euclidean algorithm is basically equivalent to the Greedy algorithm for finding the alternating Egyptian fraction representation of a rational. For instance, in your example, if we expand ...

If you know that the coefficients are non-negative and also integral, then the polynomial can be completely determined by the values of $p(1)$ and $p(p(1))$. There might be a way to extend this to ...

Chapter 4 of "Topology Now!" by Messer & Straffin gives a good undergraduate level overview of the topic of gluing polyhedral solids.

For a mind bending example, there are sequences of rationals that converge both p-adically and in the real sense to rational numbers, but not the same rational number.

12288/1663 is the 7th convergent in the infinite continued fraction representation of $e^2$, so it naturally will be a very good rational approximation. For completeness, here's the list of the first ...

The tent map. In your case it'd be scaled to $f(x) = 2 \min(x, 6-x)$.

I've been really happy with Smith, Eggen and St. Andre: http://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025/ Though that breaks your price requirement.

Possibly this is helpful: Extended Pigeonhole Principle: If $nk+1$ objects are placed in $n$ boxes, then one of the boxes must contain at least $k+1$ objects. Combine this with the probabilistic ...