Ternary parallelogram scales are MOS substitution: Difference between revisions
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Statement: If ''m'' > 1, ''n'' > 2, ''a'' has order > ''n'' in {{nowrap|ℤ/''mn''ℤ}}, and {{nowrap|ℤ/''mn''ℤ}} is partitioned into bins where one bin consists of ≤ 2''m'' - 1 adjacent elements and the rest of the bins consist of 2''m'' - 1, then {{nowrap|{0, ''a'', 2''a'', ..., (''n'' - 1)''a''}}} meets some bin at least twice. | Statement: If ''m'' > 1, ''n'' > 2, ''a'' has order > ''n'' in {{nowrap|ℤ/''mn''ℤ}}, and {{nowrap|ℤ/''mn''ℤ}} is partitioned into bins where one bin consists of ≤ 2''m'' - 1 adjacent elements and the rest of the bins consist of 2''m'' - 1, then {{nowrap|{0, ''a'', 2''a'', ..., (''n'' - 1)''a''}}} meets some bin at least twice. | ||
Proof: Apply the pigeonhole principle: | Proof: Apply the pigeonhole principle: the number of bins is | ||
<math> | <math> | ||