Löschian number

In number theory, the numbers of the form x2 + xy + y2 for integer x, y are called the Löschian numbers (or Loeschian numbers). These numbers are named after August Lösch. They are the norms of the Eisenstein integers. They are a set of whole numbers, including zero, and having prime factorization in which all primes congruent to 2 mod 3 have even powers (there is no restriction of primes congruent to 0 or 1 mod 3).

Properties

 * Every Löschian number is nonnegative.
 * Every square number is a Löschian number (by setting x or y to 0).
 * Moreover, every number of the form $$(m^2+m+1)x^2$$ for m and x integers is a Löschian number (by setting y=mx).
 * There are infinitely many Löschian numbers.
 * Given that odd and even integers are equally numerous, the probability that a Löschian number is odd is 0.75, and the probability that it is even is 0.25. This follows from the fact that $$ (x^2 + xy + y^2) $$ is even only if x and y are both even.
 * The greatest common divisor and the least common multiple of any two or more Löschian numbers are also Löschian numbers.
 * The product of two Löschian numbers is always a Löschian number, in other words Löschian numbers are closed under multiplication.
 * This property makes the set of Löschian numbers into a semigroup under multiplication.
 * The product of a Löschian number and a non-Löschian number is never a Löschian number.