Aeryk
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Taking a theorem as a definition and proving the original definition as a theorem
1 votes

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 $...

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Second Differences of Primes Pattern
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 $p_{n+2}-p_{n+1}-1.$ ...

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Any known particularities/theorems about the serie 2^a_0 + 3*2^a_1 + ... + 3^n * 2^a_n?
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)

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A particular Diophantine approximation of $\pi/2$
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}...

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Proof of the Reidemeister theorem
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 ...

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Conjectured integral for Catalan's constant
Accepted answer
10 votes

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

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Examples of theorems with proofs that have dramatically improved over time
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 ...

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How do you find maximal orders in quaternion algebras?
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 ...

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What is known about a^2 + b^2 = c^2 + d^2
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 ...

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Ring with three binary operations
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 ...

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Undergraduate Topology
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 ...

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Justifying/Explaining math research in a public address
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 ...

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Proto-Euclidean algorithm
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 ...

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Application of polynomials with non-negative coefficients
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 ...

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Topological spaces made by identifying opposite faces of a cube?
3 votes

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

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Simplifying finite sum over 1/(ax+b)
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}]

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Elementary results with p-adic numbers
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.

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Numerical coincidence involving the number 1663
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 ...

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Is there a name for a formula to calculate ascending numbers to a quadratic-like sequence?
Accepted answer
2 votes

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

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Textbook recommendations for undergraduate proof-writing class
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.

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Minimal prerequisite to reading Wiles' proof of Fermat's Last Theorem
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"

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How to solve a generalization of the Coupon Collector's problem
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 ...

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Books you would like to see translated into English
7 votes

"Arithmetique Des Algebres De Quaternions" by MF Vigneras

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