A102057 Triangle of modular inverses of b (mod m) for b = 1, ..., m-1, where 0 indicates no modular inverse exists.

1, 1, 2, 1, 0, 3, 1, 3, 2, 4, 1, 0, 0, 0, 5, 1, 4, 5, 2, 3, 6, 1, 0, 3, 0, 5, 0, 7, 1, 5, 0, 7, 2, 0, 4, 8, 1, 0, 7, 0, 0, 0, 3, 0, 9, 1, 6, 4, 3, 9, 2, 8, 7, 5, 10, 1, 0, 0, 0, 5, 0, 7, 0, 0, 0, 11, 1, 7, 9, 10, 8, 11, 2, 5, 3, 4, 6, 12, 1, 0, 5, 0, 3, 0, 0, 0, 11, 0, 9, 0, 13, 1, 8, 0, 4, 0, 0, 13

Offset

1,3

Comments

For all m, a(m, 1) = 1 and a(m, m-1) = m-1. - Evgeny Kapun, Dec 20 2016

For all coprime m and b, a(m, b)*b == 1 (mod m) and a(m, a(m, b)) = b. - Evgeny Kapun, Dec 20 2016

Links

Evgeny Kapun, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Modular Inverse

Formula

From Evgeny Kapun, Dec 20 2016: (Start)

If gcd(m, b) = 1 = mx+by (Bézout's lemma), then a(m, b) == y (mod m).

If gcd(m, b) = 1, then a(m, b) == b^(phi(m)-1) == b^(psi(m)-1) (mod m), where phi(n) is A000010 and psi(n) is A002322.

If m is prime, then a(m, b) == b^(m-2) (mod m).

If m is prime, then a(m, 1) = 1 and a(m, b) == -floor(m/b)a(m, m mod b) (mod m).

(End)

Example

m\b 1 2 3 4 5 6 7 8 9 10 11
2   1
3   1 2
4   1 0 3
5   1 3 2 4
6   1 0 0 0 5
7   1 4 5 2 3 6
8   1 0 3 0 5 0 7
9   1 5 0 7 2 0 4 8
10  1 0 7 0 0 0 3 0 9
11  1 6 4 3 9 2 8 7 5 10
12  1 0 0 0 5 0 7 0 0  0 11

Mathematica

Table[ If[ !CoprimeQ[b, m], 0, PowerMod[b, -1, m]],
(* or ModularInverse[b, m] with version >= 11.1 *)
{m, 1, 15}, {b, 1, m-1}] // Flatten (* Jean-François Alcover, Nov 09 2012, after Eric W. Weisstein *)

Prog

(Haskell) [let g 1 0 x y = x; g a 0 x y = 0; g a b x y = g b (mod a b) y (x - y * div a b) in g m b 0 1 `mod` m | m <- [1..], b <- [1..m-1]] -- Evgeny Kapun, Dec 20 2016

Crossrefs

Sequence in context: A146540 A162922 A246180 * A276276 A157497 A257961

Adjacent sequences: A102054 A102055 A102056 * A102058 A102059 A102060

Keyword

nonn,tabl

Author

Eric W. Weisstein, Dec 28 2004

Status

approved