OFFSET
0,2
COMMENTS
For n > 0, a(n) is the unique number k in [1, 5^n] and congruent to 3 mod 5 such that k^4 - 6 is divisible by 5^n.
For k not divisible by 5, k is a fourth power in 5-adic field if and only if k == 1 (mod 5). If k is a fourth power in 5-adic field, then k has exactly 4 fourth-power roots.
LINKS
Wikipedia, p-adic number
FORMULA
EXAMPLE
The unique number k in [1, 5^2] and congruent to 3 modulo 5 such that k^4 - 6 is divisible by 5^2 is k = 3, so a(2) = 3.
The unique number k in [1, 5^3] and congruent to 3 modulo 5 such that k^4 - 6 is divisible by 5^3 is k = 103, so a(3) = 103.
PROG
(PARI) a(n) = lift(-sqrtn(6+O(5^n), 4) * sqrt(-1+O(5^n)))
CROSSREFS
Approximations of p-adic fourth-power roots:
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 07 2019
STATUS
approved