forked from Hardmath123/hardmath123.github.io
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathcarnival-of-mathematics-159.html
226 lines (217 loc) · 12 KB
/
carnival-of-mathematics-159.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
<!DOCTYPE html>
<html>
<head>
<link rel="stylesheet" type="text/css" href="/static/base.css"/>
<title>Carnival of Mathematics 159 - Comfortably Numbered</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
<meta charset="utf-8"/>
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no" />
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script>
MathJax.Hub.Config({
tex2jax: {inlineMath: [['($','$)']]}
});
</script>
</head>
<body>
<header id="header">
<link rel="stylesheet" href="/octicons/octicons.css">
<script src="static/cheet.js" charset="utf-8"></script>
<script src="static/main.js"></script>
<h1>
<a href="/"><span class="left-word">Comfortably</span><span class="right-word">Numbered</span></a>
</h1>
<!-- You're reading the source to my blog? What do you think?
Do you have dinner plans? -->
</header>
<article id="postcontent" class="centered">
<section>
<h2>Carnival of Mathematics 159</h2>
<h4>Friday, July 6, 2018 · 4 min read</h4>
<blockquote>
<p>Welcome to this month’s edition of the <a href="https://aperiodical.com/carnival-of-mathematics/">Aperiodical Carnival of
Mathematics</a>!
The Carnival is a monthly roundup of exciting mathematical blog posts. Last
month, it was hosted by <a href="http://www.aperiodical.com/">Paul at the
Aperiodical</a>. This month, it is my honor
to host it here at Comfortably Numbered. But first…</p>
</blockquote>
<p><strong>Let’s play a game</strong>, shall we?</p>
<p>Pick a number. Not too large, though! You’re about to do some quick math on it.
(I’ll play along with 4.)</p>
<p>Okay. Ready? Good.</p>
<p>Now take your number and square it. (4 squared is 16.) Then, add your original
number to the square. (16 plus 4 is 20.) Finally, add forty-one. (20 plus 41 is
61.)</p>
<p>And now – your result – is it prime? Ha! I thought so. (61 certainly is.)</p>
<p>This little trick is due to Euler, who pointed out in 1772 that the polynomial
($ f(x) = x^2 + x + 41$) returns prime numbers for small integers — indeed,
all nonnegative integers up to and including 39. Since then, the quest for
other such “prime-generating” polynomials has fascinated number theorists from
around the world. As a little exercise, you may try convincing yourself that
there is no <em>perfect</em> prime-generating polynomial; that is, that there will
always be at least one integer input that gives a composite output.</p>
<p>But I digress. Here is what matters: The integers ($ x $) for which ($ f(x) $)
is composite are the deviants, the rebels, the ones who refuse to play along
with Euler’s little game.</p>
<p>Forty is the first such integer.</p>
<p>One hundred fifty-nine is another.</p>
<p>Welcome to the 159th Carnival of Mathematics.</p>
<hr>
<p>The Carnival always has a special place in its heart for clever ways to teach
children various math concepts. And this one’s no exception. In <a href="http://mathmisery.com/wp/2018/06/14/set-theory-for-second-grade/">Set Theory for
Second
Grade</a>, Manan
talks about how he designed an engaging lesson on set theory (and common
multiples!) for second graders. A quote from his students: “Can you hang this
in the hall so that everyone can see the college math we did?” </p>
<blockquote>
<p>What an amazing moment — new symbols, new concepts, no problems! At this
point, I made sure to remind them that what they are learning right now is no
different from what I would teach in college. And that if today, here in
second grade they could do college math, then in third grade they can do
third grade math, in fourth grade they can do fourth grade math, and that
they can always do math! More than a few students’ faces lit up.</p>
</blockquote>
<p>In <a href="https://cameroncounts.wordpress.com/2018/06/14/british-mathematical-colloquium-days-3-and-4/">British Mathematical Colloquium, days 3 and
4</a>,
Peter Cameron recounts in excellent detail the last couple days of the
Colloquium (it reads like a mini-Carnival!). Days 1 and 2 are linked within.</p>
<blockquote>
<p>An induced subgraph of a graph is obtained by throwing away some vertices and
the edges incident with them; you are not allowed to throw away an edge
within the set of vertices you are keeping. Paul began with the general
problem: given a graph H, can you determine the structure of graphs G
containing no induced copy of H? … The answer is known in embarrassingly
few cases … Not even for a 4-cycle is the answer known!</p>
</blockquote>
<p>In <a href="https://pballew.blogspot.com/2018/06/sum-of-cubes-is-square-of-sum-and-more.html">Sum of Cubes is Square of Sum… And
More!</a>,
Pat Ballew begins with a fact that most high-schoolers are taught, and then
rather suddenly finds himself deep in a fascinating rabbit hole. (Editor’s
note: I encourage you to read the author’s <em>On This Day in Math</em> series; I
would list all thirty of the past month if I could…)</p>
<blockquote>
<p>Like many teachers at the upper level high school math classes, over the
years I’ve presented the sum of the Cubes of the natural numbers formula
above many dozens of times. Then, perhaps like many others, I would point
out how nice it is that it turns out to be the square of the nth triangular
number, a happy coincidence that would make it easier to remember. Usually
then, we would challenge them to extend the idea to fourth powers and see if
they could do the induction proof, even though there was no really nice
simplification (to my knowledge) of the sums of fourth powers.</p>
<p>But then I reread a book that has been in my library for about six years, and
realized that many of those teachers may have known a different approach to
sums of cubes equaling square of sums that I had been completely unaware of.
In case there are other teachers who somehow also didn’t know, I share my
newfound ancient knowledge.</p>
</blockquote>
<p>In <a href="http://voices.norwich.edu/daniel-mcquillan/2018/06/29/thinking-about-the-law-of-quadratic-reciprocity/">Thinking about the Law of Quadratic
Reciprocity</a>,
Dan McQuillan gives a fast-paced overview of one of my personal favorite
theorems in number theory.</p>
<blockquote>
<p>Mathematics, the way it is currently written, can be difficult to read.
Sometimes it helps to see how people think about a topic or theorem before
(or after, or during) the reading of a proper treatment or rigorous proof.
The purpose of this post is to provide such a view regarding the proof of the
famous law of quadratic reciprocity. There are many details missing, on
purpose, and the hope is that it reads like a good story that’s both
interesting, believable and easily verifiable.</p>
</blockquote>
<hr>
<p><em>Just for fun!</em></p>
<p>In <a href="https://www.johndcook.com/blog/2018/06/16/magical-learning/">Magical
Learning</a>, John
Cook reports the results from an informal Twitter poll he conducted: “If a
genie offered to give you a thorough understanding of one theorem, what theorem
would you choose?”</p>
<p>In case you missed it, Christian at the Aperiodical is running <a href="http://aperiodical.com/2018/06/announcing-the-big-internet-math-off/">The Big
Internet
Math-Off</a>.
It may not have the intense moment-by-moment drama of the World Cup, but the
daily tidbits of math are definitely worth subscribing for.</p>
<p>In <a href="https://mathwithbaddrawings.com/2018/06/20/math-explained-through-anagrams-2/">Math Explained through
Anagrams</a>,
Ben Orlin constructs a frankly impressive amount of anagrams for various parts
of math. And they’re all illustrated! (Editor’s note: I also enjoyed <a href="https://mathwithbaddrawings.com/2018/06/27/powers-great-and-small/">this
piece</a> by
the same author. I don’t want to spoil it, but here is a wonderful quotation:
“I sometimes think that there are no puddles in math;” says Orlin, “there are
only oceans in disguise.”)</p>
<hr>
<p>That’s all I have for you this month! Come back next time when Robin at
<a href="http://www.theoremoftheday.org/">Theorem of the Day</a> will host the 160th
Carnival of Mathematics!</p>
</section>
<div id="comment-breaker">◊ ◊ ◊</div>
<!--
<section>
<center>(<a id="comment-toggle" href="#comment-toggle">Click to load comments…</a>)</center>
<div id="disqus_thread"></div>
<script type="text/javascript">
var disqus_shortname = 'comfortablynumbered';
var disqus_identifier = 'carnival-of-mathematics-159.md';
window.addEventListener(
'load',
(function() {
document.getElementById('comment-toggle').addEventListener('click',
(function() {
var dsq = document.createElement('script'); dsq.type = 'text/javascript'; dsq.async = true;
dsq.src = '//' + disqus_shortname + '.disqus.com/embed.js';
(document.getElementsByTagName('head')[0] || document.getElementsByTagName('body')[0]).appendChild(dsq);
}),
true);
}),
true
);
</script>
</section>
-->
</article>
<footer id="footer">
<div>
<ul>
<li><a href="http://github.com/Hardmath123">
<span class="octicon octicon-mark-github"></span>
Github</a></li>
<li><a href="feed.xml">
<span class="octicon octicon-rss"></span>
Subscribe (RSS)</a></li>
<li><a href="https://github.com/Hardmath123/hardmath123.github.io/issues/new?title=Hi&body=How's%20it%20going%3F">
<span class="octicon octicon-comment-discussion"></span>
Feedback</a></li>
<li><a href="mailto:contact-at-comfortablynumbered-dot-appspotmail-dot-com">
<span class="octicon octicon-mail-read"></span>
Email me</a></li>
<li><a href="http://creativecommons.org/licenses/by-nc/3.0/deed.en_US">
<span class="octicon octicon-law"></span>
CC BY-NC 3.0</a></li>
<li><a href="appreciation.html">
<span class="octicon octicon-beer"></span>
Other</a></li>
</ul>
</div>
<!--
<span class="sc">ComfortablyNumbered</span> · Hardmath123 (2013) · <a href="feed.xml">RSS Feed</a>
·
<a href="/appreciation.html">
<span class="octicon octicon-beer"></span></a>
<a rel="license" href="http://creativecommons.org/licenses/by-nc/3.0/deed.en_US">
<img
alt="Creative Commons License"
src="http://i.creativecommons.org/l/by-nc/3.0/80x15.png"/>
</a>
-->
<script>
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
ga('create', 'UA-46120535-1', 'hardmath123.github.io');
ga('require', 'displayfeatures');
ga('send', 'pageview');
</script>
</footer>
</body>
</html>