OFFSET
1,1
COMMENTS
Z_p[i] is a field iff p is a prime number congruent to 3 modulo 4.
a(n) is the number of generators of the multiplicative group Z_p[i] \ {0} (where p denotes A002145(n)).
LINKS
Rémy Sigrist, Scatterplot of (x, y) such that #{(x+i*y)^k, k >= 0} = p^2-1 (with p = A002145(62) = 647)
Rémy Sigrist, C++ program
StackExchange, Z_p[i] is a field?
EXAMPLE
For n = 2:
- the second prime number congruent to 3 modulo 4 is p = 7,
- the number of elements of {(x + i*y)^k, k >= 0} where x and y belong to Z_7 are:
x\y | 0 1 2 3 4 5 6
----+--------------------------
0 | 2 4 12 12 12 12 4
1 | 1 24 48 48 48 48 24
2 | 3 48 8 16 16 8 48
3 | 6 48 16 24 24 16 48
4 | 3 48 16 24 24 16 48
5 | 6 48 8 16 16 8 48
6 | 2 24 48 48 48 48 24
- the number 48 appears 16 times, so a(2) = 16.
PROG
(C++) // See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jun 24 2024
STATUS
approved