In number theory, the numbers of the form \( {\displaystyle x^{2}+xy+y^{2} } \) 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 square number is a Löschian number (by setting x or y to 0).
Moreover, every number of the form \( {\displaystyle (m^{2}+m+1)x^{2}} \) 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 \( {\displaystyle (x^{2}+xy+y^{2})} \) 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.
The product of a Löschian number and a non-Löschian number is never a Löschian number.
References
Marshall, J. U. (1975). "The Loeschian numbers as a problem in number theory". Geographical Analysis. 7 (4): 421–426. doi:10.1111/j.1538-4632.1975.tb01054.x.
"A003136". On-Line Encyclopedia of Integer Sequences. Retrieved 19 July 2018.
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