A318960 One of the two successive approximations up to 2^n for 2-adic integer sqrt(-7). This is the 1 (mod 4) case.

1, 5, 5, 21, 53, 53, 181, 181, 181, 181, 181, 181, 181, 16565, 49333, 49333, 49333, 49333, 573621, 1622197, 1622197, 1622197, 10010805, 10010805, 10010805, 77119669, 211337397, 479772853, 479772853, 479772853, 2627256501, 6922223797, 15512158389, 15512158389

Offset: 2

Author

Jianing Song, Sep 06 2018

Keywords

• nonn

Comments

a(n) is the unique number k in [1, 2^n] and congruent to 1 (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 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.

Examples

			The unique number k in [1, 4] and congruent to 1 modulo 4 such that k^2 + 7 is divisible by 8 is 1, so a(2) = 1.
a(2)^2 + 7 = 8 which is not divisible by 16, so a(3) = a(2) + 2^2 = 5.
a(3)^2 + 7 = 32 which is divisible by 32, so a(4) = a(3) = 5.
a(4)^2 + 7 = 32 which is divisible by 64, so a(5) = a(4) + 2^4 = 21.
a(5)^2 + 7 = 448 which is divisible by 128, so a(6) = a(5) + 2^5 = 53.
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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.

Crossrefs

Cf. A318962.

Expansions of p-adic integers:

this sequence, A318961 (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).

Programs

PARI

a(n) = truncate(-sqrt(-7+O(2^(n+1))))

Formula

a(2) = 1; 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 - A318961(n).

a(n) = Sum_{i=0..n-1} A318962(i)*2^i.

Extensions

Offset corrected by Jianing Song, Aug 28 2019