Primes of the form $X^{3} + NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

Let $n$ be an integer such that $n = 5$ or \geq 7$. In this article, we introduce a recipe for certain infinite family of non-singular plane curves degree which violate the local-global principle. Moreover, each contains infinitely many members are not geometrically isomorphic to other. Our construction is based on two arithmetic objects; is, prime numbers form $X^{3}+NY^{3}$ due Heath-Brown and Moroz Fermat type equation $x^{3}+Ny^{3} Lz^{n}$, where $N$ $L$ suitably chosen integers. sense, our extension odd was previously found by Shimizu author. The previous works only if given has divisor satisfies indivisibility conjecture Ankeny-Artin-Chowla-Mordell type. time, focus complementary cases, namely cases even degrees exceptional degrees. Consequently, well as whole. This means can unconditionally obtain every 7$ gives conclusion classical story searching explicit ternary forms violating principle, originated Selmer (1951) extended Fujiwara (1972) others.