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I think this is worth posting here, mostly because I really enjoy the simplicity of this proof but also because I have no idea how well it is known. The result is not deep or important, so the main interest is in the simplicity of the argument. Erdős proved a lower bound on the number of primes before an integer $n$. Wacław Sierpiński, in his Elementary ...

14

Here is an exposition of the proof that made Erdos famous by David Galvin. An elementary proof of Bertrand's postulate, which states that there is a prime number in between every $n$ and $2n$. The essence of this proof is in noticing that the lower bound of $$\binom{2n}{n} \geq \frac{4^n}{2n + 1}$$ The binomial expression is the middle term (and the largest)...

10

One very simple, and yet one of my favorites is the Erdős-Anning theorem: Let $A \subseteq \mathbb C$ be an infinite set of points, such that $$\forall x, y\in A \quad |x-y| \in \mathbb N$$ then there exists some $c,k \in \mathbb C$, such that all $a \in A$ is of the form $a = cx + k$ for some $x \in \mathbb R$. It was proved in 1945 in the ...

9

Erdos' proof of Infinite primes The following proof is taken from the book - "Proofs from THE BOOK" by Martin Aigner and Gunter Ziegler. This proof is attributed to Erdos. This proves that there are infinitely many primes and that the series of the sum of prime reciprocal steps diverges. Let us assume that the infinite series $\sum\frac{1}{p}$, where $p$ ...

8

Erdos' favourite questions. The following are not research papers but popular questions Erdos used to ask children. If $n+1$ integers are chosen from the first $2n$ integers, there will always be two that are co prime. There will be two numbers that are consecutive. These two numbers will be relatively prime. To see that this is not true when $n$ ...

7

Spaces in which countable intersections of open sets are open are called $P$-spaces. (Warning: the same term is also used with a completely different meaning.) The co-countable topology on an uncountable set is an example of a non-discrete $T_1$ $P$-space. In general we can start with any space $\langle X,\tau\rangle$ and let $\tau'$ be the collection of $G_\... 5$1-2\left|x-\frac12\right|$has the three-cycle$2/7,4/7,6/7$. This example comes from the paper "The Sharkovsky Theorem: A Natural Direct Proof", by Keith Burns and Boris Hasselblatt. A preprint is freely available online, and it's a great read. 5 Just pull some function values out of a hat -- for example $$f(0) = 42 \qquad f(42)=117 \qquad f(117)=0$$ and then do Lagrange interpolation (or for that matter linear interpolation, whatever floats your boat) between those points. Your attempt with a first-degree polynomial failed because a first-degree polynomial iterated three times is still a first-... 5 Erdos answered the following question in the affirmative - Are there infinitely many odd numbers that are not expressible as the sum of a prime number and a power of$2$. The proof is explained in this paper : http://www.maa.org/sites/default/files/3004416309960.pdf.bannered.pdf The essence of the proof is in showing that for every integral value of$k$, ... 4 One example is what is called a fusion system. A fusion system is a category where the objects are the subgroups of some fixed$p$-group$S$and where the morphisms is a subset of the set of injective homomorphisms between the subgroups which contains all those induced by conjugation by some element from$S$. Further, it is required that any morphism$\...

3

A good candidate would be elliptic curve cryptography. This is a direct practical application of finite fields, number theory, and other arithmetic geometry, that you would otherwise think have no purpose outside of pure mathematics.

3

Almost always still allows for an infinite yet uncountable amount to get through see Cantor Set for an example on how to build such sets. Another more trivial example would be the number is not an integer or the number is not of this "particular" countable set whatever that set may be. Other things that come to mind are things like Zigmondys theorem ...

3

Bruce Sagan's "The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions" is probably exactly what you are looking for. It covers basic representation theory but quickly moves into the representation theory of the symmetric group.

2

I think Lectures and Exercises on Functional Analysis by A. Y. Helemskii might be exactly what you're looking for. Quoting from the introduction: Perhaps the main idea is that our book is written from the categorical point of view. Everywhere we stress and comment on the categorical nature of the fundamental constructions and results (like the ...

2

I would actually go with the quaternions $\mathbb{H}$. They form a 4 dimensional, associative division algebra. With the basis $i, j , k, 1$ which satisfies $$i^2 = j^2 = k^2 = ijk = -1$$ They first might seem not useful at all, until you notice they are easily created with matrices, what means they are easily computable. Quaternions are used to compute ...

2

Here's one I like: almost all numbers fail to be computable. In particular, there are only countably many finite, terminating algorithms, and consequently there are only countably many real numbers that can be expressed (to arbitrary precision) via such algorithms. One example of a non-computable number is Chaitin's constant.

2

I like the Mathieu Groupoid $M_{13}$. Also, for any group $G$ acting on a set $X$ there is the action groupoid $X/\!/G$ with objects the elements of $X$ and with morphisms $x_1 \to x_2$ given by the elements of $G$ such that $g \cdot x_1 =x_2$. (Of course, if $G$ and $X$ are finite, so is $X/\!/G$.) (Maybe these don't quite count, as any groupoid is ...

2

Erdos - Mordell Inequality : For a point $O$ inside a given triangle $ABC$, the perpendiculars $OP$, $OQ$ and $OR$ are drawn to the side $$OA + OB + OC \geq 2(OP + OQ + OR)$$ Here's a proof from Donat K. Kazarinoff http://projecteuclid.org/download/pdf_1/euclid.mmj/1028988998 A simple visual proof is provided by Claudi Alsina and Roger Nelson in this ...

2

The proof of the Littlewood-Offord lemma for sums of real numbers. Erdos noticed that under the correspondence between sequences of $\pm$ signs and finite sets, Sperner's theorem applies and gives the optimal bound for the Littlewood-Offord lemma in dimension $1$ (on how many signed sums of $n$ given numbers of absolute value at least $1$, can have absolute ...

1

For a series of positive terms of the form $$\sum_{n=1}^{\infty}\frac{1}{n^p}$$ a. it converges if $p>1$ b. it diverges if $p\le 1$ (p-series test) For eg.-$$\sum_{n=1}^{\infty}\frac{2n+3}{n^2+5}$$ diverges as for $n\rightarrow\infty$, $u_n=\frac{2n+3}{n^2+5}\approx\frac{2n}{n^2}\approx\frac{1}{n}$.

1

Here is a neat test that is relatively unknown. In any case I can think of where it would be practical to apply, I prefer a direct comparison or limit comparison, but it's certainly still interesting and useful. Consider the series $\sum_{n=0}^\infty a_n$. Suppose you have a sequence $\{b_n\}_{n=0}^\infty$ such that $\sum_{n=0}^\infty 1/b_n$ diverges. Then ...

1

The product of consecutive integers is never a power This was proved by Erdos and Selfridge. http://www.renyi.hu/~p_erdos/1975-46.pdf

1

Perhaps a bit obscure -- finite topological spaces applied to digital analysis Perhaps a bit advanced -- spectral sequences applied to physics

1

Solution 6 There are $n^2$ ordered pairs of computers. In $n$ of these pairs, both members are the same computer. All others come in pairs with reversed order. Thus there are $\frac{n^2-n}2=\frac{n(n-1)}2$ unordered pairs of different computers. Solution 7 Put the computers in a circle and count the number of connections that a computer has to computers ...

1

When trying to prove that $$\sum^{j}_{k = i} (-1)^{j + k} \binom{j}{k} \binom{k}{i} = \delta_{i j}$$ (which is related to this porblem ) one can start with $x$ and expand the expression by adding 0, with the goal of getting two binomial coefficents. \begin{align*} x^{j} &= (x + 0)^j = (x +1 -1)^j =((x +1 ) -1 )^{j} \\ &= \sum^{j}_{k = 0} \...

1

Trolling Euclid by Tom Wright is my favorite book about unsolved problems. Very nice read.

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I saw this proof in an extract of the College Mathematics Journal. Consider the Integeral : $I$ = $\int_0^{\pi/2}ln(2cosx)dx$ From $2\cos(x)$ = $e^{ix}$ + $e^{-ix}$ , we have: $\int_0^{\pi/2}In(e^{ix}$ + $e^{-ix})dx$ = $\int_0^{\pi/2}In(e^{ix}(1 + e^{-2ix}))dx$ =$\int_0^{\pi/2}ixdx$ + $\int_0^{\pi/2}In(1 + e^{-2ix})dx$ The Taylor series expansion of ...

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