Patent val/Properties: Difference between revisions

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This page shows some properties of [[patent val]]s as well as [[generalized patent val]]s (GPVs), although currently it is all about the latter.

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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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== Sorting property ==

Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV ''V''''k'' and its next GPV ''V''''k'' + 1 are the same, and for the different entry, the latter increments the former by 1.

Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV ''V''''k'' and its next GPV {{nowrap|''V''''k'' + 1}} are the same, and for the different entry, the latter increments the former by 1.

This property states that, for example, if it is known that {{val| 12 19 28 }} is a GPV, then the next GPV is one of {{val| 13 19 28 }}, {{val| 12 20 28 }}, or {{val| 12 19 29 }}.

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{{Proof

| title=Proof

| contents=

| contents=By definition, the ''p''-limit GPV of ''n''-edo is {{nowrap|''V''(''n'') {{=}} round(''n'' log2(''Q''))}}, where ''Q'' is the prime basis {{val| 2 3 5 … ''p'' }}.

By definition, the ''p''-limit GPV of ''n''-edo is ''V'' (''n'') ~~=~~ round (''n'' log2(''Q'')), where ''Q'' is the prime basis {{val| 2 3 5 … ''p'' }}.

To prove the sorting property, we need to prove

# ~~for~~ any prime ''q''''i'' in ''Q'', there is a point of ''n'' at which ''v''''i'' gets an increment; and

# For any prime ''q''''i'' in ''Q'', there is a point of ''n'' at which ''v''''i'' gets an increment; and

# ~~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;

# 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 ''x''→''x''0+ ''f''(''x'') ~~=~~ 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~~</nowiki>~~ 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 ''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 ''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 {{w|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 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.

}}

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Given a finite prime limit, the above properties offer a way to iterate through all GPVs. To roll forwards:

# Enter a GPV. Set ''i'' = 1.

# Enter a GPV. Set {{nowrap|''i'' {{=}} 1}}.

# Copy the input and increase its ''i''-th entry by 1.

# Test if it is a GPV.

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To roll backwards:

# Enter a GPV. Set ''i'' = 1.

# Enter a GPV. Set {{nowrap|''i'' {{=}} 1}}.

# Copy the input and decrease its ''i''-th entry by 1.

# Test if it is a GPV.

@@ Line 1: / Line 1: @@
 {{Breadcrumb}}
 This page shows some properties of [[patent val]]s as well as [[generalized patent val]]s (GPVs), although currently it is all about the latter.
@@ Line 16: / Line 15: @@
 \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>π (''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>&pi;(''p'')</sub>. Find their {{w|intersection (set theory)|intersection}} ''N'', that is,
 <math> \displaystyle
@@ Line 27: / Line 26: @@
 == Sorting property ==
-Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV ''V''<sub>''k''</sub> and its next GPV ''V''<sub>''k'' + 1</sub> are the same, and for the different entry, the latter increments the former by 1.
+Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV ''V''<sub>''k''</sub> and its next GPV {{nowrap|''V''<sub>''k'' + 1</sub>}} are the same, and for the different entry, the latter increments the former by 1.
 This property states that, for example, if it is known that {{val| 12 19 28 }} is a GPV, then the next GPV is one of {{val| 13 19 28 }}, {{val| 12 20 28 }}, or {{val| 12 19 29 }}.
@@ Line 35: / Line 34: @@
 {{Proof
 | title=Proof
-| contents=
+| contents=By definition, the ''p''-limit GPV of ''n''-edo is {{nowrap|''V''(''n'') {{=}} round(''n'' log<sub>2</sub>(''Q''))}}, where ''Q'' is the prime basis {{val| 2 3 5 … ''p'' }}.
-By definition, the ''p''-limit GPV of ''n''-edo is ''V'' (''n'') &#61; round (''n'' log<sub>2</sub>(''Q'')), where ''Q'' is the prime basis {{val| 2 3 5 … ''p'' }}.
 To prove the sorting property, we need to prove
-# for any prime ''q''<sub>''i''</sub> in ''Q'', there is a point of ''n'' at which ''v''<sub>''i''</sub> gets an increment; and
+# For any prime ''q''<sub>''i''</sub> in ''Q'', there is a point of ''n'' at which ''v''<sub>''i''</sub> gets an increment; and
-# 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 ''x''→''x''<sub>0</sub><sup>+</sup> ''f''(''x'') &#61; lim ''x''→ ''x''<sub>0</sub><sup>-</sup> ''f''(''x'') + 1.
+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}}.
-<nowiki>#1</nowiki> holds immediately following the definition of the round function, and the point is ''n'' &#61; (''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 ''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 ''y'' &#61; 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>)) &#61; 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 <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.
 }}
@@ Line 53: / Line 50: @@
 Given a finite prime limit, the above properties offer a way to iterate through all GPVs. To roll forwards:
-# Enter a GPV. Set ''i'' = 1.
+# Enter a GPV. Set {{nowrap|''i'' {{=}} 1}}.
 # Copy the input and increase its ''i''-th entry by 1.
 # Test if it is a GPV.
@@ Line 60: / Line 57: @@
 To roll backwards:
-# Enter a GPV. Set ''i'' = 1.
+# Enter a GPV. Set {{nowrap|''i'' {{=}} 1}}.
 # Copy the input and decrease its ''i''-th entry by 1.
 # Test if it is a GPV.