# 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.

• How about finding integer solutions to equations of the form $x^2+y^2=z^2$ ? – Weltschmerz Jun 18 '12 at 19:07
• Pell's equation (en.wikipedia.org/wiki/Pell's_equation) – user17762 Jun 18 '12 at 19:13
• Investigate (and prove?) divisibility properties of Fibonacci numbers? – Old John Jun 18 '12 at 19:15
• I have flagged this to be made into a community wiki. – user17762 Jun 18 '12 at 19:17
• This might be too broad, but multiplicative functions and interesting/important. – The Chaz 2.0 Jun 18 '12 at 19:34

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:

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

• The proof is trivial if we proceed by induction. – Anunay Kulshrestha Jun 18 '12 at 20:08
• No proof is trivial enough, but, good for you. =) – Pedro Tamaroff Jun 18 '12 at 20:09
• @AnunayKulshrestha While that's true, you can still look at the derivation or intuition of choosing such a formula in the first place. Proceeding by induction once the formula to prove has been given is only running the second half of the race. – Robert Mastragostino Jun 18 '12 at 20:10
• @RobertMastragostino Maybe I shouldn't have given the result away, that'd have been more interesting. – Pedro Tamaroff Jun 18 '12 at 20:11
• @PedroTamaroff, regarding your statement under Perfect numbers: "Euler showed that if a number is perfect, then it is of Euclid's kind." Of course, this assumes that there are no odd perfect numbers? Because otherwise, it is currently unknown if an odd perfect number can be triangular. (See e.g. this MO question and the answers contained therein.) It is possible to show though, that an odd perfect number is a nontrivial multiple of the triangular number $T(q) = q(q+1)/2$, where $q$ is the Euler prime of the odd perfect number $N = q^k n^2$. – Jose Arnaldo Bebita-Dris Sep 9 '17 at 15:15

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

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

• Wikipedia is a good reference. – The Chaz 2.0 Jun 18 '12 at 19:33
• I'm looking for open problems, not the history of Number Theory. Anyway, thank you. – Anunay Kulshrestha Jun 18 '12 at 19:34
• @AnunayKulshrestha Open problems? You didn't state that in your question. – Pedro Tamaroff Jun 18 '12 at 19:37
• @AnunayKulshrestha You do realize there's a reason they're called open right? =) One needs a very vast amount of machinery before tackling any of those. It is great you're interested, but I think it is more down-to-earth to start with the basics. ;) – Pedro Tamaroff Jun 18 '12 at 19:47
• If you want open problems, take a look here: garden.irmacs.sfu.ca – Integral Jun 18 '12 at 19:51