|
|
A322563
|
|
One of the two successive approximations up to 17^n for 17-adic integer sqrt(-2). This is the 7 (mod 17) case (except for n = 0).
|
|
5
|
|
|
0, 7, 24, 3492, 3492, 755181, 755181, 386956285, 3669665669, 38548452874, 1935954476826, 30159869083112, 30159869083112, 612782106312873, 149181452599901928, 2001337544753312147, 47800106368910364835, 777717984503913392050, 7395640079594607505466
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
For n > 0, a(n) is the unique solution to x^2 == -2 (mod 17^n) in the range [0, 17^n - 1] and congruent to 7 modulo 17.
A322564 is the approximation (congruent to 10 mod 17) of another square root of -2 over the 17-adic field.
|
|
LINKS
|
|
|
FORMULA
|
For n > 0, a(n) = 17^n - A322564(n).
a(n) = Sum_{i=0..n-1} A322565(i)*17^i.
|
|
EXAMPLE
|
7^2 = 49 = 3*17 - 2;
24^2 = 576 = 2*17^2 - 2;
3492^2 = 12194064 = 2482*17^3 - 2.
|
|
MATHEMATICA
|
{0}~Join~Table[First@Select[PowerModList[-2, 1/2, 17^k], Mod[#, 17]==7&], {k, 20}] (* Giorgos Kalogeropoulos, Sep 14 2022 *)
|
|
PROG
|
(PARI) a(n) = truncate(sqrt(-2+O(17^n)))
|
|
CROSSREFS
|
Approximations of 17-adic square roots:
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|