Patent val/Properties: Difference between revisions

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\frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>

Denote the solution sets as ''N''1, ''N''2, …, ''N''~~π~~(''p''). Find their {{w|intersection (set theory)|intersection}} ''N'', that is,

Denote the solution sets as ''N''1, ''N''2, …, ''N''π(''p''). Find their {{w|intersection (set theory)|intersection}} ''N'', that is,

<math> \displaystyle

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# For any distinct primes ''q''''i'', ''q''''j'' in ''Q'', there is ''not'' a point of ''n'' at which both ''v''''i'' and ''v''''j'' get an increment;

where an increment of ''f''(''x'') at ''x''0 is defined as lim {{nowrap|''x'' ~~→~~ ''x''0+ ''f''(''x'')}} {{=}} {{nowrap|lim ''x'' ~~→~~ ''x''0− ''f''(''x'') + 1}}.

where an increment of ''f''(''x'') at ''x''0 is defined as lim {{nowrap|''x'' → ''x''0+ ''f''(''x'')}} {{=}} {{nowrap|lim ''x'' → ''x''0− ''f''(''x'') + 1}}.

<nowiki />#1 holds immediately following the definition of the round function, and the point is ''n'' {{=}}(''v''''i'' + 1/2)/log2(''q''''i'').

<nowiki/>#1 holds immediately following the definition of the round function, and the point is {{nowrap|''n'' {{=}} (''v''''i'' + 1/2)/log2(''q''''i'')}}.

To prove ~~<nowiki>~~#2~~</nowiki>~~, 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''''i''|''q''''j'' ∈ ''Q''|2''n'' log2(''q''''i'') ∈ '''Z'''|~~and~~ 2''n'' log2(''q''''j'') ∈ '''Z'''}}. If that is the case, then their quotient {{nowrap|(2''n'' log2(''q''''i''))/(2''n'' log2(''q''''j'')) {{=}} log''q''''j''(''q''''i'') ∈ '''Q'''}}, which contradicts {{w|Gelfond–Schneider theorem}}. Therefore, the hypothesis is false, and such an ''n'' does not exist.

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''''i'', ''q''''j'' ∈ ''Q''}}, {{nowrap|2''n'' log2(''q''''i'') ∈ '''Z'''}}, and {{nowrap|2''n'' log2(''q''''j'') ∈ '''Z'''}}. If that is the case, then their quotient {{nowrap|(2''n'' log2(''q''''i''))/(2''n'' log2(''q''''j'')) {{=}} log''q''''j''(''q''''i'') ∈ '''Q'''}}, which contradicts {{w|Gelfond–Schneider theorem}}. Therefore, the hypothesis is false, and such an ''n'' does not exist.

}}

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 \frac {v_{\pi (q)} - 1/2}{\log_2 (q)} < n < \frac {v_{\pi (q)} + 1/2}{\log_2 (q)} </math>
-Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>&pi;(''p'')</sub>. Find their {{w|intersection (set theory)|intersection}} ''N'', that is,
+Denote the solution sets as ''N''<sub>1</sub>, ''N''<sub>2</sub>, …, ''N''<sub>π(''p'')</sub>. Find their {{w|intersection (set theory)|intersection}} ''N'', that is,
 <math> \displaystyle
@@ Line 40: / Line 40: @@
 # 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'' &rarr; ''x''<sub>0</sub><sup>+</sup> ''f''(''x'')}} {{=}} {{nowrap|lim ''x'' &rarr; ''x''<sub>0</sub><sup>&minus;</sup> ''f''(''x'') + 1}}.
+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>&minus;</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 <nowiki>#2</nowiki>, 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''|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 {{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.
+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.
 }}