Löschian number explained
| Named After: | August Lösch |
| Parentsequence: | A032766 |
| Formula: | x2 + xy + y2 for integer x, y |
| First Terms: | 0, 1, 3, 4, 7, 9, 12, 13, 16 |
| Oeis: | A003136 |
| Oeis Name: | Löschian numbers |
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
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
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.
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