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).

Löschian number

Named after	August Lösch
Subsequence of	A032766
Formula	x2 + xy + y2 for integer x, y
First terms	0, 1, 3, 4, 7, 9, 12, 13, 16
OEIS index	A003136 Löschian numbers

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 monoid 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.