Ternary scale theorems: Difference between revisions
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Proof: Write '''u''' and '''v''' for the two sizes of ''j''-steps. Since gcd(''j'', 2''b'') = 1, there exists ''m'' such that stacking ''m''-many ''j''-steps yields scale steps of ''U'', and ''m'' is odd because gcd(''m'', 2''b'') = 1. Hence the scale steps of ''U'' are ('''uv''')<sup>(''m''−1)/2</sup>'''u''' mod '''e''' and ('''vu''')<sup>(''m''−1)/2</sup>'''v''' mod '''e''', and the step sizes alternate because '''u''' and '''v''' do. | Proof: Write '''u''' and '''v''' for the two sizes of ''j''-steps. Since gcd(''j'', 2''b'') = 1, there exists ''m'' such that stacking ''m''-many ''j''-steps yields scale steps of ''U'', and ''m'' is odd because gcd(''m'', 2''b'') = 1. Hence the scale steps of ''U'' are ('''uv''')<sup>(''m''−1)/2</sup>'''u''' mod '''e''' and ('''vu''')<sup>(''m''−1)/2</sup>'''v''' mod '''e''', and the step sizes alternate because '''u''' and '''v''' do. | ||
These two claims prove that ''E''<sub>'''X'''</sub>(S) = ('''YZ''')<sup>''b''</sup> and that the two GS generators' sizes differ by replacing one '''Y''' for a '''Z'''. | These two claims prove that ''E''<sub>'''X'''</sub>(S) = ('''YZ''')<sup>''b''</sup> and that the two GS generators' sizes differ by replacing one '''Y''' for a '''Z'''. {{Qed}} | ||
== Theorem 2 (Odd generator-offset scales are SGA) == | == Theorem 2 (Odd generator-offset scales are SGA) == | ||