Patent val/Properties: Difference between revisions
ArrowHead294 (talk | contribs) m Look at the template and the associated module—«nowrap» with multiple arguments only makes the text in the arguments fixed; by default, the comma and spaces separating the text in the arguments are not non-breaking Tag: Undo |
m Undo revision 167678 by ArrowHead294 (talk). There's no point in using a template argument separator where it's supposed to be an ordinary comma (punctuation), be it breaking or nonbreaking Tag: Undo |
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# For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment; | # For any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in ''Q'', there is ''not'' a point of ''n'' at which both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> get an increment; | ||
where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap|''x'' | where an increment of ''f''(''x'') at ''x''<sub>0</sub> is defined as lim {{nowrap|''x'' → ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap|lim ''x'' → ''x''<sub>0</sub><sup>−</sup> ''f''(''x'') + 1}}. | ||
<nowiki/>#1 holds immediately following the definition of the round function, and the point is ''n'' {{=}}(''v''<sub>''i''</sub> + 1/2)/log<sub>2</sub>(''q''<sub>''i''</sub>). | <nowiki/>#1 holds immediately following the definition of the round function, and the point is {{nowrap|''n'' {{=}} (''v''<sub>''i''</sub> + 1/2)/log<sub>2</sub>(''q''<sub>''i''</sub>)}}. | ||
To prove | To prove #2, let us assume there exists such an ''n''. By the definition of the round function, an increment of {{nowrap|''y'' {{=}} round(''x'')}} occurs only if {{nowrap|2''x'' ∈ '''Z'''}}. Thus, for any distinct primes {{nowrap|''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> ∈ ''Q''}}, {{nowrap|2''n'' log<sub>2</sub>(''q''<sub>''i''</sub>) ∈ '''Z'''}}, and {{nowrap|2''n'' log<sub>2</sub>(''q''<sub>''j''</sub>) ∈ '''Z'''}}. If that is the case, then their quotient {{nowrap|(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 {{w|Gelfond–Schneider theorem}}. Therefore, the hypothesis is false, and such an ''n'' does not exist. | ||
}} | }} | ||