A Positive-Rank Elliptic Curve with No Dense Prime

An AIM problem asks whether every positive-rank elliptic curve over \(\mathbb Q\) has a prime \(p\) for which its rational points are dense in its \(p\) -adic points. We give a negative answer. For \[ E:y^2=x^3-1516563, \qquad P=(6403/9,511280/27), \] an unconditional full \(2\) -descent and saturation certify \(E(\mathbb Q)=\mathbb ZP\) . A rational \(3\) -isogeny places the reduction of every rational point in a subgroup of index \(3\) at every good prime, while separate exact arguments exclude density at the three bad primes \(2,3,79\) . Thus the dense-prime set is empty. We also record an exact finite local criterion: density is equivalent to surjectivity on a fixed finite quotient of the Néron filtration, and at a good odd prime on \(\mathcal E(\mathbb Z/p^2\mathbb Z)\) .