Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 625221961 - Original comment: ** |
Wikispaces>TallKite **Imported revision 625289603 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-24 02:12:11 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>625289603</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n') | Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n') | ||
Can b' be reduced by simplifying further? | Can b' be reduced by simplifying further? | ||
Let r = GCD (a', b') = GCD (q - ap, -bp) | |||
GCD (q - ap, p) = GCD (q, p) = 1 | |||
Therefore r <= b, and r = GCD (q - ap, b) | |||
b | [//unfinished proof//] | ||
To prove: alternate true/false test | To prove: alternate true/false test | ||
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(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n') | (P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n') | ||
Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b | Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b | ||
[//needs more work//] | |||
|b'| = m, so the unreduced pergen is explicitly false, and the test works | |b'| = m, so the unreduced pergen is explicitly false, and the test works | ||
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Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n')<br /> | Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n')<br /> | ||
Can b' be reduced by simplifying further?<br /> | Can b' be reduced by simplifying further?<br /> | ||
Let r = GCD (a', b') = GCD (q - ap, -bp)<br /> | |||
GCD (q - ap, p) = GCD (q, p) = 1<br /> | |||
Therefore r &lt;= b, and r = GCD (q - ap, b)<br /> | |||
[<em>unfinished proof</em>]<br /> | |||
<br /> | <br /> | ||
To prove: alternate true/false test<br /> | To prove: alternate true/false test<br /> | ||
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(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')<br /> | (P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')<br /> | ||
Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b<br /> | Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b<br /> | ||
[<em>needs more work</em>]<br /> | |||
|b'| = m, so the unreduced pergen is explicitly false, and the test works<br /> | |b'| = m, so the unreduced pergen is explicitly false, and the test works<br /> | ||
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