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A318961 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 3 (mod 4) case. 3
3, 3, 11, 11, 11, 75, 75, 331, 843, 1867, 3915, 8011, 16203, 16203, 16203, 81739, 212811, 474955, 474955, 474955, 2572107, 6766411, 6766411, 23543627, 57098059, 57098059, 57098059, 57098059, 593968971, 1667710795, 1667710795, 1667710795, 1667710795, 18847579979 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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).

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 start from the rightmost digit for odd primes p, while for p = 2 they differ start from the second rightmost digit. As a result, the formulas and the program here are different from those in other entries related to p-adic integers.

LINKS

Jianing Song, Table of n, a(n) for n = 2..999 (offset corrected by Jianing Song)

G. P. Michon, Introduction to p-adic integers, Numericana.

FORMULA

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).

a(n) = 2^n - A318960(n).

a(n) = Sum_{i=0..n-1} A318963(i)*2^i.

EXAMPLE

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.

a(2)^2 + 7 = 16 which is divisible by 16, so a(3) = a(2) = 3.

a(3)^2 + 7 = 16 which is not divisible by 32, so a(4) = a(3) + 2^3 = 11.

a(4)^2 + 7 = 128 which is divisible by 64, so a(5) = a(4) = 11.

a(5)^2 + 7 = 128 which is divisible by 128, so a(6) = a(5) = 11.

...

PROG

(PARI) a(n) = if(n==2, 3, truncate(sqrt(-7+O(2^(n+1)))))

CROSSREFS

Cf. A318963.

Expansions of p-adic integers:

A318960, this sequence (2-adic, sqrt(-7));

A268924, A271222 (3-adic, sqrt(-2));

A268922, A269590 (5-adic, sqrt(-4));

A048898, A048899 (5-adic, sqrt(-1));

A290567 (5-adic, 2^(1/3));

A290568 (5-adic, 3^(1/3));

A290800, A290802 (7-adic, sqrt(-6));

A290806, A290809 (7-adic, sqrt(-5));

A290803, A290804 (7-adic, sqrt(-3));

A210852, A212153 (7-adic, (1+sqrt(-3))/2);

A290557, A290559 (7-adic, sqrt(2));

A286840, A286841 (13-adic, sqrt(-1));

A286877, A286878 (17-adic, sqrt(-1)).

Also expansions of 10-adic integers:

A007185, A010690 (nontrivial roots to x^2-x);

A216092, A216093, A224473, A224474 (nontrivial roots to x^3-x).

Sequence in context: A095019 A321082 A167428 * A309798 A068594 A147175

Adjacent sequences:  A318958 A318959 A318960 * A318962 A318963 A318964

KEYWORD

nonn

AUTHOR

Jianing Song, Sep 06 2018

EXTENSIONS

Offset corrected by Jianing Song, Aug 28 2019

STATUS

approved

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Last modified September 27 13:13 EDT 2020. Contains 337380 sequences. (Running on oeis4.)