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 = − ^{1}⁄_{2} + 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:
< a_{1}, b_{1} > < a_{2}, b_{2} > = < a_{1}a_{2} − b_{1}b_{2}, a_{1}b_{2} + b_{1}a_{2} − b_{1}b_{2} >
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 < A_{8}, B_{8} > to represent whatever < a, b >^{8} might be. The object is find A_{8} and B_{8}.
To form stage one, write the following terms as candidates for A_{8} and B_{8}. The coefficients come from Pascal's triangle, and the superscripts follow an obvious pattern. At this stage, A_{8} and B_{8} have the same candidates.
stage one | ||
---|---|---|
candidates for A_{8} | candidates for B_{8} | |
1 a^{8} b^{0} | 1 a^{8} b^{0} | |
8 a^{7} b^{1} | 8 a^{7} b^{1} | |
28 a^{6} b^{2} | 28 a^{6} b^{2} | |
56 a^{5} b^{3} | 56 a^{5} b^{3} | |
70 a^{4} b^{4} | 70 a^{4} b^{4} | |
56 a^{3} b^{5} | 56 a^{3} b^{5} | |
28 a^{2} b^{6} | 28 a^{2} b^{6} | |
8 a^{1} b^{7} | 8 a^{1} b^{7} | |
1 a^{0} b^{8} | 1 a^{0} b^{8} |
At stage two:
stage two | ||||
---|---|---|---|---|
candidates for A_{8} | candidates for B_{8} | |||
1 a^{8} b^{0} | 1 a^{8} b^{0} | delete | ||
8 a^{7} b^{1} | delete | 8 a^{7} b^{1} | ||
28 a^{6} b^{2} | 28 a^{6} b^{2} | |||
56 a^{5} b^{3} | 56 a^{5} b^{3} | delete | ||
70 a^{4} b^{4} | delete | 70 a^{4} b^{4} | ||
56 a^{3} b^{5} | 56 a^{3} b^{5} | |||
28 a^{2} b^{6} | 28 a^{2} b^{6} | delete | ||
8 a^{1} b^{7} | delete | 8 a^{1} b^{7} | ||
1 a^{0} b^{8} | 1 a^{0} b^{8} |
For stage three, assign alternating signs to the remaining candidates for each of A_{8} and B_{8}, starting with positive:
stage three | ||
---|---|---|
candidates for A_{8} | candidates for B_{8} | |
+1 a^{8} b^{0} | ||
+8 a^{7} b^{1} | ||
−28 a^{6} b^{2} | −28 a^{6} b^{2} | |
+56 a^{5} b^{3} | ||
+70 a^{4} b^{4} | ||
−56 a^{3} b^{5} | −56 a^{3} b^{5} | |
+28 a^{2} b^{6} | ||
+8 a^{1} b^{7} | ||
−1 a^{0} b^{8} | −1 a^{0} b^{8} |
At stage four, gather the remaining terms:
stage four | ||||||
---|---|---|---|---|---|---|
A_{8} = | 1a^{8}b^{0} | − 28a^{6}b^{2} | + 56a^{5}b^{3} | − 56a^{3}b^{5} | + 28a^{2}b^{6} | − 1a^{0}b^{8} |
B_{8} = | 8a^{7}b^{1} | − 28a^{6}b^{2} | + 70a^{4}b^{4} | − 56a^{3}b^{5} | + 8a^{1}b^{7} | − 1a^{0}b^{8} |
Now < A_{8}, B_{8} > is determined.
Here are formulas for specific powers. If the power p is a multiple of three, A_{p} will have one more term than B_{p}.
< a, b >^{2} = < A_{2}, B_{2} > | ||
A_{2} = | 1a^{2}b^{0} | − 1a^{0}b^{2} |
B_{2} = | 2a^{1}b^{1} | − 1a^{0}b^{2} |
< a, b >^{3} = < A_{3}, B_{3} > | |||
A_{3} = | 1a^{3}b^{0} | − 3a^{1}b^{2} | + 1a^{0}b^{3} |
B_{3} = | 3a^{2}b^{1} | − 3a^{1}b^{2} |
< a, b >^{4} = < A_{4}, B_{4} > | |||
A_{4} = | 1a^{4}b^{0} | − 6a^{2}b^{2} | + 4a^{1}b^{3} |
B_{4} = | 4a^{3}b^{1} | − 6a^{2}b^{2} | + 1a^{0}b^{4} |
< a, b >^{5} = < A_{5}, B_{5} > | ||||
A_{5} = | 1a^{5}b^{0} | − 10a^{3}b^{2} | + 10a^{2}b^{3} | − 1a^{0}b^{5} |
B_{5} = | 5a^{4}b^{1} | − 10a^{3}b^{2} | + 5a^{1}b^{4} | − 1a^{0}b^{5} |
< a, b >^{6} = < A_{6}, B_{6} > | |||||
A_{6} = | 1a^{6}b^{0} | − 15a^{4}b^{2} | + 20a^{3}b^{3} | − 6a^{1}b^{5} | + 1a^{0}b^{6} |
B_{6} = | 6a^{5}b^{1} | − 15a^{4}b^{2} | + 15a^{2}b^{4} | − 6a^{1}b^{5} |
< a, b >^{7} = < A_{7}, B_{7} > | |||||
A_{7} = | 1a^{7}b^{0} | − 21a^{5}b^{2} | + 35a^{4}b^{3} | − 21a^{2}b^{5} | + 7a^{1}b^{6} |
B_{7} = | 7a^{6}b^{1} | − 21a^{5}b^{2} | + 35a^{3}b^{4} | − 21a^{2}b^{5} | + 1a^{0}b^{7} |