# Number TheoryUnsolved problems in number theory

#### matqkks

##### Member
What are the most interesting examples of unsolved problems in number theory which an 18 year can understand?

#### chisigma

##### Well-known member
What are the most interesting examples of unsolved problems in number theory which an 18 year can understand?
With the term 'twin primes' are defined pairs of prime numbers $p_{1},p{2}$ where $p_{2}= p_{1} + 2$... examples are $5-7$, $11-13$, $17-19$ and so one... the general opinion is that the 'twin primes' are infinite but nobody till now has demonstrated that... may be that one of Your students will meet this remarkable goal!...

Kind regards

$\chi$ $\sigma$

#### chisigma

##### Well-known member
Another famous unsolved number theory problem is also related to the prime numbers: the Goldbach's Conjecture...

Goldbach Conjecture -- from Wolfram MathWorld

... and it is fully ubderstable also for kidds...

Kind regards

$\chi$ $\sigma$

#### ModusPonens

##### Well-known member
Wasn't the Goldbach conjecture proved recently?

#### chisigma

##### Well-known member
Wasn't the Goldbach conjecture proved recently?
Actually there are two 'Goldback's Conjectures', the Goldback's weak conjecture originally proposed by Goldback in a famous letter sent to Euler in 1742 and that extablishes that...

Every integer greater than 5 can be written as the sum of three primes

... and the Goldback's strong conjecture that extablishes that...

Every even integer greater than 2 can be written as the sum of two primes

In 2013 the Peruvian mathematician Herald Helfgott released two papers claiming a proof of the Goldback's weak conjecture...

Kind regards

$\chi$ $\sigma$

#### mathbalarka

##### Well-known member
MHB Math Helper
The Birch & Swinnerton-Dyer conjecture is one of my favourites, although that belongs to Analytic Number Theory, a much broader branch of general NT.

EDIT -- A short introduction, I thought, would be nice, so here it is :

The main conjecture is that rank of any elliptic curve over any global field is equal to it's order of the zero of the L-function $$\displaystyle L(E, s)$$ at s = 1. The rank can explicitly be determined in terms of the period, regulator and the order of Tate-Shafarevich group.

Did the above made sense? Perhaps another equivalent statement may be described would be helpfull (much like a consequence of it) :

N is the area of a right triangle with rational sides if an only if the number of multisets over $$\displaystyle \mathbb{Z},$$ $$\displaystyle (x, y, z)$$, such that $$\displaystyle 2x^2 + y^2 + 8z^2 = N$$ with z odd is equal to the number of multisets over $$\displaystyle \mathbb{Z}$$ satisfying the same equation with z even.

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#### eddybob123

##### Active member
There are many sites that are made specially for the purpose of posting unsolved problems in mathematics. One of them is Open Problem Garden, which you will find a distinct variety of them. Usually, the ones with low importance are the simpler ones.