Patent val/Properties: Difference between revisions

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<nowiki>#1</nowiki> holds because the point is ''N'' = (''v''''i'' + 1/2)/log2 (''q''''i'').

To prove <nowiki>#2</nowiki>, let us assume there exists such an ''N''. By the property of the round function, an increment of ''y'' = round (''x'') occurs ~~iff~~ 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''''i'', ''q''''j'' in q, 2''N'' log2 (''q''''i'') ∈ '''Z''', and 2''N'' log2 (''q''''j'') ∈ '''Z'''. If that is the case, then their quotient (2''N'' log2 (''q''''i''))/(2''N'' log2 (''q''''j'')) = log''q''''j'' (''q''''i'') ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''N'' does not exist.

To prove <nowiki>#2</nowiki>, let us assume there exists such an ''N''. By the property of the round function, an increment of ''y'' = round (''x'') occurs only if 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''''i'', ''q''''j'' in q, 2''N'' log2 (''q''''i'') ∈ '''Z''', and 2''N'' log2 (''q''''j'') ∈ '''Z'''. If that is the case, then their quotient (2''N'' log2 (''q''''i''))/(2''N'' log2 (''q''''j'')) = log''q''''j'' (''q''''i'') ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''N'' does not exist.

== Application ==

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 <nowiki>#1</nowiki> holds because the point is ''N'' = (''v''<sub>''i''</sub> + 1/2)/log<sub>2</sub> (''q''<sub>''i''</sub>).
-To prove <nowiki>#2</nowiki>, let us assume there exists such an ''N''. By the property of the round function, an increment of ''y'' = round (''x'') occurs iff 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in q, 2''N'' log<sub>2</sub> (''q''<sub>''i''</sub>) ∈ '''Z''', and 2''N'' log<sub>2</sub> (''q''<sub>''j''</sub>) ∈ '''Z'''. If that is the case, then their quotient (2''N'' log<sub>2</sub> (''q''<sub>''i''</sub>))/(2''N'' log<sub>2</sub> (''q''<sub>''j''</sub>)) = log<sub>''q''<sub>''j''</sub></sub> (''q''<sub>''i''</sub>) ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''N'' does not exist.
+To prove <nowiki>#2</nowiki>, let us assume there exists such an ''N''. By the property of the round function, an increment of ''y'' = round (''x'') occurs only if 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in q, 2''N'' log<sub>2</sub> (''q''<sub>''i''</sub>) ∈ '''Z''', and 2''N'' log<sub>2</sub> (''q''<sub>''j''</sub>) ∈ '''Z'''. If that is the case, then their quotient (2''N'' log<sub>2</sub> (''q''<sub>''i''</sub>))/(2''N'' log<sub>2</sub> (''q''<sub>''j''</sub>)) = log<sub>''q''<sub>''j''</sub></sub> (''q''<sub>''i''</sub>) ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''N'' does not exist.
 == Application ==