Generator-offset property: Difference between revisions
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By modular arithmetic we have ''rk'' ≡ ''k''/2 mod ''n'' iff ''r'' ≡ (''n'' + 1)/2 mod ''n''. (Note that both 2 and ''k'' are coprime with ''n'', hence multiplicatively invertible mod ''n''.) This proves that the offset, which must be reached after (''n'' + 1)/2 ''k''-steps, is a ''k''/2-step, as desired. (If the offset wasn't reached in (''n'' + 1)/2 steps, the two generator chains either wouldn't be disjoint or wouldn't have the assumed lengths.) | By modular arithmetic we have ''rk'' ≡ ''k''/2 mod ''n'' iff ''r'' ≡ (''n'' + 1)/2 mod ''n''. (Note that both 2 and ''k'' are coprime with ''n'', hence multiplicatively invertible mod ''n''.) This proves that the offset, which must be reached after (''n'' + 1)/2 ''k''-steps, is a ''k''/2-step, as desired. (If the offset wasn't reached in (''n'' + 1)/2 steps, the two generator chains either wouldn't be disjoint or wouldn't have the assumed lengths.) | ||
=== Proposition 3 (Properties of even generator-offset scales) === | === Proposition 3 (Properties of even generator-offset ternary scales) === | ||
A primitive generator-offset scale of even size where the generator g is an even-step (i.e. g subtends an even number of steps) has the following properties: | A primitive generator-offset ternary scale of even size where the generator g is an even-step (i.e. g subtends an even number of steps) has the following properties: | ||
# It is a union of two primitive mosses of size ''n''/2 generated by g | # It is a union of two primitive mosses of size ''n''/2 generated by g | ||
# It is ''not'' SV3 | # It is ''not'' SV3 | ||