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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.

  • 0
    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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    @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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    *to be honest I'm very ske$p$tical about the article I put i$n$ 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