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Suborder Function


The multiplicative suborder of a number a (mod n) is the least exponent e>0 such that a^e=+/-1 (mod n), or zero if no such e exists. An e always exists if GCD(a,n)=1 and n>1.

This function is denoted sord_n(a) and can be implemented in the Wolfram Language as:

  Suborder[a_,n_] := If[n>1&& GCD[a,n] == 1,
    Min[MultiplicativeOrder[a, n, {-1, 1}]],
    0
  ]

The following table summarizes sord_n(a) for small values of a and n.

aOEISsord_n(a) for n=0, 1, ...
20, 0, 0, 1, 0, 2, 0, 3, 0, 3, 0, 5, 0, 6, 0, ...
3A1034890, 0, 1, 0, 1, 2, 0, 3, 2, 0, 2, 5, 0, 3, 3, ...
40, 0, 0, 1, 0, 1, 0, 3, 0, 3, 0, 5, 0, 3, 0, ...
5A1034910, 0, 1, 1, 1, 0, 1, 3, 2, 3, 0, 5, 2, 2, 3, ...

See also

Multiplicative Order

This entry contributed by Tony Noe

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequences A103489 and A103491 in "The On-Line Encyclopedia of Integer Sequences."Wolfram, S.; Martin, O.; and Odlyzko, A. M. "Algebraic Properties of Cellular Automata." Comm. Math. Phys. 93, 219-258, 1984.

Referenced on Wolfram|Alpha

Suborder Function

Cite this as:

Noe, Tony. "Suborder Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SuborderFunction.html

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