OFFSET
0,2
COMMENTS
For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 2 mod 13 such that k^4 - 3 is divisible by 13^n.
For k not divisible by 13, k is a fourth power in 13-adic field if and only if k == 1, 3, 9 (mod 13). If k is a fourth power in 13-adic field, then k has exactly 4 fourth-power roots.
LINKS
Wikipedia, p-adic number
FORMULA
EXAMPLE
The unique number k in [1, 13^2] and congruent to 2 modulo 13 such that k^4 - 3 is divisible by 13^2 is k = 28, so a(2) = 28.
The unique number k in [1, 13^3] and congruent to 2 modulo 13 such that k^4 - 3 is divisible by 13^3 is k = 1211, so a(3) = 1211.
PROG
(PARI) a(n) = lift(sqrtn(3+O(13^n), 4) * sqrt(-1+O(13^n)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 01 2019
STATUS
approved