I'm looking for some mathematics that will challenge me as a year $12$ student. [closed]

I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.

I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number $1$ favorite thing and I really want something to keep me busy and something that can further my understanding of mathematics. Also I would be interested in any mathematical focused book suggestions.

So far in school I've done the usual:

Matrices, transformation matrices, Sine Cosine and Tangent (graphs and proofs), lots and lots of parabolas/quadratics, statistics, growth and decay, calculus intro, Calculus derivation and integration, vectors, proof by induction and complex numbers.

Any suggestions would be heavily appreciated.

Solution 1:

I have been asked for a hint. Gauss was thinking about roots of unity. The method of Gauss for dealing with these polynomials is in chapter 9 of Cox Galois Theory. In fact, as the author points out, this work predates Galois Theory by about thirty years. To give a name, although it will not help with these problems, https://en.wikipedia.org/wiki/Gaussian_period

For the indicated prime in the simplest problem below, define $$ \omega = e^{2 \pi i / p} = \cos \frac{2 \pi }{p} + i \sin \frac{2 \pi }{p}$$ Then $$ \omega^p = 1 $$ and the indicated real root, when a single cosine term, is just $$ \omega + \frac{1}{\omega} = \omega + \omega^{p-1}$$

For the problems with two cosine terms, the indicated root is, with some integer $k,$ $$ \omega + \omega^k + \omega^{p-k} + \omega^{p-1}$$

Show that $$ x = 2 \cos \left( \frac{2 \pi}{7} \right) $$ is a root of $$ x^3 + x^2 - 2 x - 1. $$ For this one, find all the roots. This appears on page 6 of Reuschle

Show that $$ x = 2 \cos \left( \frac{2 \pi}{11} \right) $$ is a root of $$ x^5 + x^4 -4 x^3 -3 x^2 + 3 x + 1. $$ This appears on page 9 of Reuschle

Show that $$ x = 2 \cos \left( \frac{2 \pi}{23} \right) $$ is a root of $$ x^{11} + x^{10} - 10 x^9 - 9 x^8 + 36 x^7 + 28 x^6 - 56 x^5 - 35 x^4 + 35 x^3 + 15 x^2 - 6 x - 1. $$ This one appears on page 30 of Reuschle

Show that $$ x = 2 \cos \left( \frac{2 \pi}{47} \right) $$ is a root of $$ x^{23} + x^{22} - 22 x^{21} - 21 x^{20} + 210 x^{19} + 190 x^{18} -1140 x^{17} -969 x^{16} + 3876 x^{15} + 3060 x^{14} $$ $$ -8568 x^{13} - 6188 x^{12} + 12376 x^{11} + 8008 x^{10} - 11440 x^9 - 6435 x^8 + 6435 x^7 + 3003 x^6 - 2002 x^5 $$ $$ -715 x^4 + 286 x^3 + 66 x^2 - 12 x - 1 $$ This one appears on page 73 of Reuschle. Really nice. I have ordered a cheap paperback reprint of Reuschle.

Note that $$ 2, 5, 11, 23, 47 $$ are the maximal chain of Sophie Germain primes (well, at least when they are not $4 \pmod 5$); apparently $47$ is called a "safe prime" instead. In any string of integers $x_1,x_2,x_3,x_4,x_5,$ such that $x_{n+1} = 2 x_n + 1,$ one of the string is divisible by $5.$ The five numbers can only be primes if one of them is equal to $5.$ Meanwhile, $ x = 2 \cos \left( \frac{2 \pi}{47} \right) $ is a root of $x^2 + x - 1,$ which begins the chain of Gaussian minimal polynomals.

We can get a longer chain when the first one is $-1 \pmod {30},$ i.e. $$ 89, \; 179, \; 359, \; 719, \; 1439, \; 2879. $$ $$ 1122659, \; 2245319, \; 4490639, \; 8981279, \; 17962559, \; 35925119, \; 71850239. $$

Find at least one root of $$ x^3 + x^2 - 4 x + 1. $$ Sum of a pair of cosines this time, denominator $13$. It is actually quite unusual to have one of these where the "main" root is a single cosine term. That happens only when the degree is prime $q$, while $2q+1$ is also prime, while the polynomial is constructed very carefully; the recipe is due to Gauss. Section seven in the Disquisitiones Arithmeticae.

Find at least one root of $$ x^7 + x^6 - 12 x^5 - 7 x^4 + 28 x^3 + 14 x^2 - 9 x + 1. $$ Sum of a pair of cosines this time, denominator $29$. This one appears on page 35 of Reuschle

Solution 2:

Project Euler is a great source of interesting problems. Many of them require you to learn a little computer programming, which I highly recommend you try if you haven't before. (And if you don't have a preferred programming language, give Sage a try. Nice clean syntax with an extensive math library and you don't even have to install anything.)

Just for a sense of flavor, here's an early problem about the Collatz Conjecture that I rather like. It's slow by brute force, but a bit of recursive magic solves it in under a second.

The following iterative sequence is defined for the set of positive integers:

$n \rightarrow n/2$ ($n$ is even)

$n \rightarrow 3n + 1$ ($n$ is odd)

Using the rule above and starting with $13$, we generate the following sequence:

$13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1$

It can be seen that this sequence (starting at $13$ and finishing at $1$) contains $10$ terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at $1$.

Which starting number, under one million, produces the longest chain?

Solution 3:

If you are really seeking for recomandation I can suggest you some books:

$1$.Putnam and beyond: Book by Razvan Gelca and Titu Andreescu.

$2$. Elementary Number Theory: Primes, Congruences, and Secrets:By Willaim Stein.

$3$.Mathematical Diamonds:By Ross Honsberger.

These three books are the best collection for developing strong logical skills and mastering the problem solving abilities. Other's opinion can be different from that of mine but since I m also an upcoming year $12$ student so I thought you will like what I like.