A324027 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 2 (mod 5) case (except for n = 0).

0, 2, 12, 37, 162, 1412, 10787, 42037, 354537, 1526412, 3479537, 3479537, 3479537, 247620162, 3909729537, 10013245162, 101565979537, 711917542037, 2237796448287, 13681888245162, 51828860901412, 337931155823287, 1291605472229537, 10828348636292037, 58512064456604537

Offset: 0

Author

Jianing Song, Sep 07 2019

Keywords

• nonn

Comments

For n > 0, a(n) is the unique solution to x^2 == -6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 2 modulo 5.

A324028 is the approximation (congruent to 3 mod 5) of another square root of 6 over the 5-adic field.

Examples

			12^2 = 144 = 6*5^2 - 6;
37^2 = 1369 = 11*5^3 - 6;
162^2 = 26244 = 42*5^4 - 6.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A048898, A048899, A324029, A324030.

Approximations of 5-adic square roots:

this sequence, A324028 (sqrt(-6));

A268922, A269590 (sqrt(-4));

A048898, A048899 (sqrt(-1));

A324023, A324024 (sqrt(6)).

Programs

PARI

a(n) = truncate(sqrt(-6+O(5^n)))

Formula

For n > 0, a(n) = 5^n - A324028(n).

a(n) = A048898(n)*A324023(n) mod 5^n = A048899(n)*A324024(n) mod 5^n.