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- Feb 13, 2012

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

Kind regards

$\chi$ $\sigma$

- Feb 13, 2012

- 1,704

Goldbach Conjecture -- from Wolfram MathWorld

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

Kind regards

$\chi$ $\sigma$

- Jun 26, 2012

- 45

Wasn't the Goldbach conjecture proved recently?

- Feb 13, 2012

- 1,704

Actually there are two 'Goldback's Conjectures', theWasn't the Goldbach conjecture proved recently?

... and the

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

Kind regards

$\chi$ $\sigma$

- Mar 22, 2013

- 573

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.

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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- Aug 18, 2013

- 76