Powers of Eisenstein integers.
Version of Monday 7 February 2022.
Dave Barber's other pages.

Eisenstein integers are a subset of the complex numbers, and are akin to the Gaussian integers.

An Eisenstein integer can be written as an ordered pair of integers, the components of which are often named a and b. In this report, the two components of an Eisenstein integer are enclosed in angle brackets: < a, b >. By contrast, a complex number in rectangular coordinates is written in square brackets: [ x, y ].

A convenient constant is R = √0.75. An essential constant for Eisenstein integers is ω = − 12 + R. The fundamental definition:

< a, b > = a + bω

< a, b > can be converted into the ordinary complex representation [ x, y ] with these formulas:

 x = a − b ÷ 2 y = b × R

Many properties of Eisenstein integers survive if a and b are not limited to the integers, but are instead allowed to be any real numbers. In that case, the conversion can be inverted:

 a = x + y ÷ R ÷ 2 b = y ÷ R

The multiplication rule is a consequence of the definitions above:

< a1, b1 > < a2, b2 > = < a1a2b1b2, a1b2 + b1a2b1b2 >

The object of this report is to display the pattern formed when Eisenstein integers are raised to integer powers. Because a general formula is unwieldy, a procedure involving a specific example is given; the eighth power is chosen because it has enough substance to reveal what happens.

Write < A8, B8 > to represent whatever < a, b >8 might be. The object is find A8 and B8.

To form stage one, write the following terms as candidates for A8 and B8. The coefficients come from Pascal's triangle, and the superscripts follow an obvious pattern. At this stage, A8 and B8 have the same candidates.

stage one
candidates
for A8
candidates
for B8
1 a8 b0 1 a8 b0
8 a7 b1 8 a7 b1
28 a6 b2 28 a6 b2
56 a5 b3 56 a5 b3
70 a4 b4 70 a4 b4
56 a3 b5 56 a3 b5
28 a2 b6 28 a2 b6
8 a1 b7 8 a1 b7
1 a0 b8 1 a0 b8

At stage two:

• delete the second candidate for A8, and every third candidate thereafter;
• delete the first candidate for B8, and every third candidate thereafter.

stage two
candidates
for A8
candidates
for B8
1 a8 b0 1 a8 b0delete
8 a7 b1delete 8 a7 b1
28 a6 b2 28 a6 b2
56 a5 b3 56 a5 b3delete
70 a4 b4delete 70 a4 b4
56 a3 b5 56 a3 b5
28 a2 b6 28 a2 b6delete
8 a1 b7delete 8 a1 b7
1 a0 b8 1 a0 b8

For stage three, assign alternating signs to the remaining candidates for each of A8 and B8, starting with positive:

stage three
candidates
for A8
candidates
for B8
+1 a8 b0
+8 a7 b1
−28 a6 b2 −28 a6 b2
+56 a5 b3
+70 a4 b4
−56 a3 b5 −56 a3 b5
+28 a2 b6
+8 a1 b7
−1 a0 b8 −1 a0 b8

At stage four, gather the remaining terms:

stage four
A8 = 1a8b0 − 28a6b2 + 56a5b3 − 56a3b5 + 28a2b6 − 1a0b8
B8 = 8a7b1 − 28a6b2 + 70a4b4 − 56a3b5 + 8a1b7 − 1a0b8

Now < A8, B8 > is determined.

Here are formulas for specific powers. If the power p is a multiple of three, Ap will have one more term than Bp.

 < a, b >2 = < A2, B2 > A2 = 1a2b0 − 1a0b2 B2 = 2a1b1 − 1a0b2

 < a, b >3 = < A3, B3 > A3 = 1a3b0 − 3a1b2 + 1a0b3 B3 = 3a2b1 − 3a1b2

 < a, b >4 = < A4, B4 > A4 = 1a4b0 − 6a2b2 + 4a1b3 B4 = 4a3b1 − 6a2b2 + 1a0b4

 < a, b >5 = < A5, B5 > A5 = 1a5b0 − 10a3b2 + 10a2b3 − 1a0b5 B5 = 5a4b1 − 10a3b2 + 5a1b4 − 1a0b5

 < a, b >6 = < A6, B6 > A6 = 1a6b0 − 15a4b2 + 20a3b3 − 6a1b5 + 1a0b6 B6 = 6a5b1 − 15a4b2 + 15a2b4 − 6a1b5

 < a, b >7 = < A7, B7 > A7 = 1a7b0 − 21a5b2 + 35a4b3 − 21a2b5 + 7a1b6 B7 = 7a6b1 − 21a5b2 + 35a3b4 − 21a2b5 + 1a0b7