cp's OEIS Frontend

A318961 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 3 (mod 4) case.

%I A318961 #44 Dec 17 2021 08:26:45
%S A318961 3,3,11,11,11,75,75,331,843,1867,3915,8011,16203,16203,16203,81739,
%T A318961 212811,474955,474955,474955,2572107,6766411,6766411,23543627,
%U A318961 57098059,57098059,57098059,57098059,593968971,1667710795,1667710795,1667710795,1667710795,18847579979
%N A318961 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 3 (mod 4) case.
%C A318961 a(n) is the unique number k in [1, 2^n] and congruent to 3 (mod 4) such that k^2 + 7 is divisible by 2^(n+1).
%C A318961 The 2-adic integers are very different from p-adic ones where p is an odd prime. For example, provided that there is at least one solution, the number of solutions to x^n = a over p-adic integers is gcd(n, p-1) for odd primes p and gcd(n, 2) for p = 2. For odd primes p, x^2 = a is solvable iff a is a quadratic residue modulo p, while for p = 2 it's solvable iff a == 1 (mod 8). If gcd(n, p-1) > 1 and gcd(a, p) = 1, then the solutions to x^n = a differ starting at the rightmost digit for odd primes p, while for p = 2 they differ starting at the next-to-rightmost digit. As a result, the formulas and the program here are different from those in other entries related to p-adic integers.
%H A318961 Jianing Song, <a href="/A318961/b318961.txt">Table of n, a(n) for n = 2..999</a> (offset corrected by Jianing Song)
%H A318961 G. P. Michon, <a href="http://www.numericana.com/answer/p-adic.htm#integers">Introduction to p-adic integers</a>, Numericana.
%F A318961 a(2) = 3; for n >= 3, a(n) = a(n-1) if a(n-1)^2 + 7 is divisible by 2^(n+1), otherwise a(n-1) + 2^(n-1).
%F A318961 a(n) = 2^n - A318960(n).
%F A318961 a(n) = Sum_{i=0..n-1} A318963(i)*2^i.
%e A318961 The unique number k in [1, 4] and congruent to 3 modulo 4 such that k^2 + 7 is divisible by 8 is 3, so a(2) = 3.
%e A318961 a(2)^2 + 7 = 16 which is divisible by 16, so a(3) = a(2) = 3.
%e A318961 a(3)^2 + 7 = 16 which is not divisible by 32, so a(4) = a(3) + 2^3 = 11.
%e A318961 a(4)^2 + 7 = 128 which is divisible by 64, so a(5) = a(4) = 11.
%e A318961 a(5)^2 + 7 = 128 which is divisible by 128, so a(6) = a(5) = 11.
%e A318961 ...
%o A318961 (PARI) a(n) = if(n==2, 3, truncate(sqrt(-7+O(2^(n+1)))))
%Y A318961 Cf. A318963.
%Y A318961 Expansions of p-adic integers:
%Y A318961 A318960, this sequence (2-adic, sqrt(-7));
%Y A318961 A268924, A271222 (3-adic, sqrt(-2));
%Y A318961 A268922, A269590 (5-adic, sqrt(-4));
%Y A318961 A048898, A048899 (5-adic, sqrt(-1));
%Y A318961 A290567 (5-adic, 2^(1/3));
%Y A318961 A290568 (5-adic, 3^(1/3));
%Y A318961 A290800, A290802 (7-adic, sqrt(-6));
%Y A318961 A290806, A290809 (7-adic, sqrt(-5));
%Y A318961 A290803, A290804 (7-adic, sqrt(-3));
%Y A318961 A210852, A212153 (7-adic, (1+sqrt(-3))/2);
%Y A318961 A290557, A290559 (7-adic, sqrt(2));
%Y A318961 A286840, A286841 (13-adic, sqrt(-1));
%Y A318961 A286877, A286878 (17-adic, sqrt(-1)).
%Y A318961 Also expansions of 10-adic integers:
%Y A318961 A007185, A010690 (nontrivial roots to x^2-x);
%Y A318961 A216092, A216093, A224473, A224474 (nontrivial roots to x^3-x).
%K A318961 nonn
%O A318961 2,1
%A A318961 _Jianing Song_, Sep 06 2018
%E A318961 Offset corrected by _Jianing Song_, Aug 28 2019