|
|
A325893
|
|
The successive approximations up to 2^n for 2-adic integer 5^(1/5).
|
|
4
|
|
|
0, 1, 1, 5, 5, 21, 21, 21, 149, 149, 149, 1173, 1173, 1173, 9365, 9365, 42133, 107669, 238741, 500885, 1025173, 1025173, 1025173, 1025173, 1025173, 1025173, 34579605, 34579605, 34579605, 34579605, 571450517, 1645192341, 1645192341, 1645192341, 10235126933, 10235126933
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
a(n) is the unique solution to x^5 == 5 (mod 2^n) in the range [0, 2^n - 1].
|
|
LINKS
|
|
|
FORMULA
|
For n > 0, a(n) = a(n-1) if a(n-1)^5 - 5 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).
|
|
EXAMPLE
|
For n = 2, the unique solution to x^5 == 5 (mod 4) in the range [0, 3] is x = 1, so a(2) = 1.
a(2)^5 - 5 = -4 which is not divisible by 8, so a(3) = a(2) + 4 = 5;
a(3)^5 - 5 = 3120 which is divisible by 16, so a(4) = a(3) = 5;
a(4)^5 - 5 = 3120 which is not divisible by 32, so a(5) = a(4) + 16 = 21;
a(5)^5 - 5 = 4084096 which is divisible by 64, so a(6) = a(5) = 21.
|
|
PROG
|
(PARI) a(n) = lift(sqrtn(5+O(2^n), 5))
|
|
CROSSREFS
|
For the digits of 5^(1/5), see A325897.
Approximations of p-adic fifth-power roots:
this sequence (2-adic, 5^(1/5));
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|