A322560 One of the two successive approximations up to 17^n for 17-adic integer sqrt(2). This is the 11 (mod 17) case (except for n = 0).

0, 11, 45, 623, 39927, 958658, 17996942, 66272080, 886949426, 63668766395, 1723899037353, 3739892937802, 38011789245435, 2951122975394240, 111901481337359547, 1795679746931368837, 27557487210519710974, 708814173469855869708, 2363294697242529398062

Offset: 0

Author

Jianing Song, Aug 29 2019

Keywords

• nonn

Comments

For n > 0, a(n) is the unique solution to x^2 == 2 (mod 17^n) in the range [0, 17^n - 1] and congruent to 11 modulo 17.

A322559 is the approximation (congruent to 6 mod 17) of another square root of 2 over the 17-adic field.

Examples

			11^2 = 121 = 7*17 + 2;
45^2 = 2025 = 7*17^2 + 2;
623^2 = 388129 = 79*17^3 + 2.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A322561, A322562.

Approximations of 17-adic square roots:

A286877, A286878 (sqrt(-1));

A322559, this sequence (sqrt(2));

A322563, A322564 (sqrt(-2)).

Programs

PARI

a(n) = truncate(-sqrt(2+O(17^n)))

Formula

For n > 0, a(n) = 17^n - A322559(n).

a(n) = Sum_{i=0..n-1} A322562(i)*17^i.

a(n) = A286877(n)*A322563(n) mod 17^n = A286878(n)*A322564(n) mod 17^n.