A324082 One of the four successive approximations up to 13^n for 13-adic integer 3^(1/4).This is the 3 (mod 13) case (except for n = 0).

0, 3, 68, 575, 13757, 156562, 4612078, 52880168, 178377202, 9967145854, 137221138330, 1240089073122, 22746013801566, 279024950148857, 2399150696294628, 2399150696294628, 104770936724476142, 3431853982640375347, 98586429095835092610, 1335595905567366417029

Offset: 0

Author

Jianing Song, Sep 01 2019

Keywords

• nonn

Comments

For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 3 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.

Examples

			The unique number k in [1, 13^2] and congruent to 3 modulo 13 such that k^4 - 3 is divisible by 13^2 is k = 68, so a(2) = 68.
The unique number k in [1, 13^3] and congruent to 3 modulo 13 such that k^4 - 3 is divisible by 13^3 is k = 575, so a(3) = 575.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A286840, A286841, A322085, A324077, A324083, A324084, A324085, A324086, A324087, A324153.

Programs

PARI

a(n) = lift(sqrtn(3+O(13^n), 4))

Formula

a(n) = A324077(n)*A286841(n) mod 13^n = A324084(n)*A286840(n) mod 13^n.

For n > 0, a(n) = 13^n - A324083(n).

a(n)^2 == A322086(n) (mod 13^n).