A324028 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 3 (mod 5) case (except for n = 0).

0, 3, 13, 88, 463, 1713, 4838, 36088, 36088, 426713, 6286088, 45348588, 240661088, 973082963, 2193786088, 20504332963, 51021911088, 51021911088, 1576900817338, 5391598082963, 43538570739213, 138906002379838, 1092580318786088, 1092580318786088, 1092580318786088

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 3 modulo 5.

A324027 is the approximation (congruent to 3 mod 5) of another square root of -6 over the 5-adic field.

Examples

			13^2 = 169 = 7*5^2 - 6;
88^2 = 7744 = 62*5^3 - 6;
463^2 = 214369 = 343*5^4 - 6.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A048898, A048899, A324029, A324030.

Approximations of 5-adic square roots:

A324027, sequence (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 - A324027(n).

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