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 $$L(E, s)$$ 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 $$\mathbb{Z},$$ $$(x, y, z)$$, such that $$2x^2 + y^2 + 8z^2 = N$$ with z odd is equal to the number of multisets over $$\mathbb{Z}$$ satisfying the same equation with z even.