High school mathematical research

I am a grade 12 student. I am interested in number theory and I am looking for topics to research on.

Can you suggest some topics in number theory and in general that would make for a good research project?

I have self-studied certain topics in Abstract Algebra and Number Theory. I'm fascinated by primes (like most people are).

Preferably, suggest some unexplored problems so that new results can be obtained.

Thanks.

Solution 1:

NOTE The OP didn't state "Preferably, suggest some open problems so that new results can be obtained." when this was answered.

I can provide you with Burton's Elementary Number Theory. It has a series of historical introductions and great examples you'll probably find worth of a research project. He has information and obivously theory about results from Fermat, Euler, Diophantus, Wilson, Möbius, and others. I can also provide you with the three volumes of the History of Number Theory, which might be a great source.

A few examples are

Fermat's Little Theorem If $p\not\mid a$ then$$a^{p-1} \equiv 1 \mod p$$

Wilson's Theorem If $p$ is a prime then

$$({p-1})! \equiv -1 \mod p$$

Möbius Inversion Formula If we have two arithmetical functions $f$ and $g$ such that

$$f(n) = \sum_{d \mid n} g(d)$$

Then

$$g(n) = \sum_{d \mid n} f(d)\mu\left(\frac{n}{d}\right)$$

Where $\mu$ is the Möbius function.

Maybe so interesting as the previous,

The $\tau$ and $\sigma$ functions

Let $\tau(n)$ be the number of divisors of $n$ and $\sigma(n)$ its sum. Then if $$n=p_1^{l_1}\cdots p_k^{l_k}$$

$$\tau(n)=\prod_{m=1 }^k(1+l_m)$$

$$\sigma(n)=\prod_{m=1 }^k \frac{p^{l_m+1}-1}{p-1}$$

Legendre's Identity

The multiplicty (i.e. number of times) with which $p$ divides $n!$ is

$$\nu(n)=\sum_{m=1}^\infty \left[\frac{n}{p^m} \right]$$

However odd that might look, the argument is somehow simple. The multiplicity with which $p$ divides $n$ is $\left[\dfrac{n}{p} \right]$, for $p^2$ it is $\left[\dfrac{n}{p^2} \right]$, and so forth. To get that of $n!$ we sum all these values to get the above, since each of $1,\dots,n$ is counted $l$ times as a multiple of $p^m$ for $m=1,2,\dots,l$, if $p$ divides it exactly $l$ times. Note the sum will terminate because the least integer function $[x]$ is zero when $p^m>n$.

Perfect numbers

A number is called a perfect number is the sum if its divisors equals the number, this means

$$\sigma(n) =2n$$

Euclid showed if $p=2^n-1$ is a prime, then $$\frac{p(p+1)}{2}$$ is always a perfect number

Euler showed that if a number is perfect, then it is of Euclid's kind.

$n$ - agonal or figurate numbers.

The greeks were very interested in numbers that could be decomposed into geometrical figures. The square numbers are well known to us, namely $m=n^2$. But what about triangular, or pentagonal numbers?

Explicit formulas have been found, namely

$$t_n=\frac{n(n+1)}{2}$$

$$p_n=\frac{n(3n-1)}{2}$$

You can try, as a good olympiadish excercise, to prove the following:

$${t_1} + {t_2} + {t_3} + \cdots + {t_n} = \frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{6}$$

We can arrange the numbers in a pentagon as a triangle and a square:

$${p_n} = {t_{n - 1}} + {n^2}$$

Solution 2:

Wikipedia is a good reference to see some history about number theory:

http://en.wikipedia.org/wiki/Number_theory