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 $... View answer Accepted answer 6 votes For any sequence$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$p_{n+2}-p_{n+1}-1.$... View answer 1 votes 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) View answer 4 votes Yes, such a subsequence exists. [Edit: An infinite number of] The continued fraction convergents,$\frac{p_n}{q_n}$, for$\pi/2$satisfy $$\left|\frac{\pi}{2}-\frac{p_n}{q_n}\right|<\frac{1}{2q_n^2}... View answer Accepted answer 3 votes Mathematica (or rather Wolfram Alpha) gives an answer in terms of the digamma function: http://www.wolframalpha.com/input/?i=Sum[1%2F%28a+x%2Bb%29%2C{x%2Cx0%2Cx1}] View answer 2 votes Messer and Straffin's book "Topology Now!" provides most of the steps starting from their definition of a knot (able to decomposed into a finite number of linear segments), building up the ideas of ... View answer Accepted answer 10 votes Mathematica confirms the following: Change the integral to polar coordinates to get$$\frac{(1)}{2} = \int_0^{\pi/4} \int_0^{\sec(\theta)/2} \frac{1}{1-r^2}r\ dr\ d\theta = \frac{G}{6}.$$View answer 8 votes Some of the big ideas and connections (with lots of pertinent references) are presented excellently in Fernando Q. Gouvea's "A Marvelous Proof" View answer 4 votes 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 ... View answer 5 votes 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 ... View answer 4 votes 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 ... View answer 5 votes 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 ... View answer 12 votes 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 ... View answer 1 votes 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 ... View answer 7 votes 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 ... View answer 14 votes 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 ... View answer 3 votes Chapter 4 of "Topology Now!" by Messer & Straffin gives a good undergraduate level overview of the topic of gluing polyhedral solids. View answer 14 votes 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. View answer 6 votes 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 ... View answer Accepted answer 2 votes The tent map. In your case it'd be scaled to$f(x) = 2 \min(x, 6-x)$. View answer 4 votes 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. View answer 0 votes 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 ...