Ternary scale theorems: Difference between revisions
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Suppose ''s'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''s'' projects to primitive MOSes ''s''<sub>1</sub> (via identifying '''b''' with '''c'''), ''s''<sub>2</sub> (via identifying '''a''' with '''c'''), and ''s''<sub>3</sub> (via identifying '''a''' with '''b'''). Suppose ''s''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: '''P''', the perfect ''k''-step, and '''I''', the imperfect ''k''-step. By stacking ''n''-many ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''s''<sub>2</sub> and ''s''<sub>3</sub>, respectively. These binary words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be MOSes, since ''m''-steps in the new words correspond to ''mk''-steps in the MOS words ''s''<sub>1</sub> and ''s''<sub>2</sub>, which come in at most two sizes. Since ''s''<sub>1</sub> is a primitive MOS, {{nowrap|gcd(''k'', ''n'') {{=}} 1}}. Hence when {{nowrap|0 < ''m'' < ''n''}}, ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are primitive MOSes. | Suppose ''s'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''s'' projects to primitive MOSes ''s''<sub>1</sub> (via identifying '''b''' with '''c'''), ''s''<sub>2</sub> (via identifying '''a''' with '''c'''), and ''s''<sub>3</sub> (via identifying '''a''' with '''b'''). Suppose ''s''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: '''P''', the perfect ''k''-step, and '''I''', the imperfect ''k''-step. By stacking ''n''-many ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''s''<sub>2</sub> and ''s''<sub>3</sub>, respectively. These binary words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be MOSes, since ''m''-steps in the new words correspond to ''mk''-steps in the MOS words ''s''<sub>1</sub> and ''s''<sub>2</sub>, which come in at most two sizes. Since ''s''<sub>1</sub> is a primitive MOS, {{nowrap|gcd(''k'', ''n'') {{=}} 1}}. Hence when {{nowrap|0 < ''m'' < ''n''}}, ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are primitive MOSes. | ||
<pre> | <pre<includeonly />> | ||
index: 1 2 3 4 ... ''n'' | index: 1 2 3 4 ... ''n'' | ||
Σ<sub>1</sub>: '''P P P P ... P I''' | Σ<sub>1</sub>: '''P P P P ... P I''' | ||