A103491 Multiplicative suborder of 5 (mod n) = sord(5, n).

0, 0, 1, 1, 1, 0, 1, 3, 2, 3, 0, 5, 2, 2, 3, 0, 4, 8, 3, 9, 0, 3, 5, 11, 2, 0, 2, 9, 6, 7, 0, 3, 8, 10, 8, 0, 6, 18, 9, 4, 0, 10, 3, 21, 5, 0, 11, 23, 4, 21, 0, 16, 4, 26, 9, 0, 6, 18, 7, 29, 0, 15, 3, 3, 16, 0, 10, 11, 16, 11, 0, 5, 6, 36, 18, 0, 9, 30, 4, 39, 0, 27, 10, 41, 6, 0, 21, 7, 10, 22, 0

Offset: 0

Author

Harry J. Smith, Feb 08 2005

Keywords

• easy
• nonn

Comments

a(n) is minimum e for which 5^e = +/-1 mod n, or zero if no e exists.

For n > 2, a(n) <= (n-1)/2, with equality if (but not only if) n is in A019335. - Robert Israel, Mar 20 2020

References

• H. Cohen, Course in Computational Algebraic Number Theory, Springer, 1993, p. 25, Algorithm 1.4.3

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Multiplicative Order.
• S. Wolfram, Algebraic Properties of Cellular Automata (1984), Appendix B.
• Eric Weisstein's World of Mathematics, Suborder Function

Crossrefs

Cf. A019335.

Programs

Maple

f:= proc(n) local x;
  if n mod 5 = 0 then return 0 fi;
  x:= numtheory:-mlog(-1,5,n);
  if x <> FAIL then x else numtheory:-order(5,n) fi
end proc:
f(1):= 0:
map(f, [$0..100]); # Robert Israel, Mar 20 2020

Mathematica

Suborder[k_, n_] := If[n > 1 && GCD[k, n] == 1, Min[MultiplicativeOrder[k, n, {-1, 1}]], 0];
a[n_] := Suborder[5, n];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 21 2020, after T. D. Noe in A003558 *)