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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.

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    How about finding integer solutions to equations of the form $x^2+y^2=z^2$ ?2012-06-18
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    Pell's equation (http://en.wikipedia.org/wiki/Pell's_equation)2012-06-18
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    Investigate (and prove?) divisibility properties of Fibonacci numbers?2012-06-18
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    I have flagged this to be made into a community wiki.2012-06-18
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    This might be too broad, but [multiplicative functions](http://en.wikipedia.org/wiki/Multiplicative_function) and interesting/important.2012-06-18
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    You could take a look at [this question](http://math.stackexchange.com/questions/158595/prime-as-sum-of-three-numbers-whose-product-is-a-cube)2012-06-18

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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:

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

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    The proof is trivial if we proceed by induction.2012-06-18
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    No proof is trivial enough, but, good for you. =)2012-06-18
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    @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.2012-06-18
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    @RobertMastragostino Maybe I shouldn't have given the result away, that'd have been more interesting.2012-06-18
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    @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](https://mathoverflow.net/questions/234528) 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 $N = q^k n^2$.2017-09-09
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    @PedroTamaroff, what is true however, is that Euclid's form $$E = (2^p - 1){2^{p-1}}$$ for even perfect numbers and Euler's form $$O = q^k n^2$$ for odd perfect numbers possess *very similar* multiplicative structures -- they are both of the form $$M = Q^K N^2,$$ for some prime $Q$, $K \equiv 1 \pmod 4$ and $\gcd(Q, N) = 1$.2017-09-09
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Wikipedia is a good reference to see some history about number theory:

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

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    Wikipedia is a good reference.2012-06-18
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    I'm looking for open problems, not the history of Number Theory. Anyway, thank you.2012-06-18
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    @AnunayKulshrestha **Open** problems? You didn't state that in your question.2012-06-18
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    Sorry, just added that.2012-06-18
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    Yes, that changes things, but my earlier suggestion about divisibility properties of Fibonacci numbers includes open problems, if you include "is there an infinity of Fibonacci primes".2012-06-18
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    @Anunay You must be in a very special high school. You can spend a lifetime just studying the advances made towards solutions of open problems.2012-06-18
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    @Weltschmerz I plan to major in math and try to make advances myself. :)2012-06-18
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    @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. ;)2012-06-18
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    @PeterTamaroff I'm not referring to the open problems that are well-known like for example Goldbach's Conjecture. I'm referring to unexplored areas suitable for a high school student interested in obtaining new results.2012-06-18
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    If you want open problems, take a look here: http://garden.irmacs.sfu.ca/2012-06-18
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    @AnunayKulshrestha I see. I don't know about that. Open problems in high school level wouldn't be of relevance, methinks. Doesn't any of what I wrote call your attention? What do you know about number theory?2012-06-18
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    @PeterTamaroff It does. I think I'll have to go back to the basics and try to find a good topic. I've read books like Number Theory Structures, Introduction to Number Theory etc on my own (mostly for olympiads). I don't have any formal education in the subject.2012-06-18
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    You need to be at like Master's degree level (after undergraduate college) to tackle open problems. You really shouldn't look for high school level open problems, I don't think any exist. However, there have been exceptional high school student(s) who solved open problems in high school. (See http://educationviews.org/rutvik-oza-solves-an-unsolved-problem-in-mathematics/)2013-02-19
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    *to be honest I'm very skeptical about the article I put in the link above. (See http://physics.stackexchange.com/questions/28931/what-are-the-precise-statements-by-shouryya-ray-of-particle-dynamics-problems-po#comment110293_28931) I don't know whether any high school student has actually solved any open problem before, but basically my point is that as a high school student you really shouldn't be looking for open problems to solve. (as far as I know)2013-02-19