Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 625347349 - Original comment: ** |
Wikispaces>TallKite **Imported revision 625347399 - 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>2018-01-25 03: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-25 03:55:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>625347399</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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To prove: test for explicitly false | To prove: test for explicitly false | ||
If m = |b|, is the pergen explicitly false? | If m = |b|, is the pergen explicitly false? | ||
Does (a,b)/n split P8 into | Does (a,b)/n split P8 into m periods? | ||
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5 | (a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5 | ||
Because n is a multiple of b, n/b is an integer | Because n is a multiple of b, n/b is an integer | ||
M/b = (n/b)·M/n = (n/b)·G | M/b = (n/b)·M/n = (n/b)·G | ||
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5) | (a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5) | ||
Let c and d be the bezout pair of a+b and b | Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1 | ||
Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0 | Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0 | ||
c·(a+b)·P8 = c·b·((n/b)·G - P5) | c·(a+b)·P8 = c·b·((n/b)·G - P5) | ||
(1 - d·b)·P8 = c·b·((n/b)·G - P5) | (1 - d·b)·P8 = c·b·((n/b)·G - P5) | ||
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) | P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5) | ||
P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G) | P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G) | ||
Therefore P8 is split into | Therefore P8 is split into m periods | ||
Therefore if m = |b|, the pergen is explicitly false | Therefore if m = |b|, the pergen is explicitly false | ||
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To prove: test for explicitly false<br /> | To prove: test for explicitly false<br /> | ||
If m = |b|, is the pergen explicitly false?<br /> | If m = |b|, is the pergen explicitly false?<br /> | ||
Does (a,b)/n split P8 into | Does (a,b)/n split P8 into m periods?<br /> | ||
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5<br /> | (a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5<br /> | ||
Because n is a multiple of b, n/b is an integer<br /> | Because n is a multiple of b, n/b is an integer<br /> | ||
M/b = (n/b)·M/n = (n/b)·G<br /> | M/b = (n/b)·M/n = (n/b)·G<br /> | ||
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)<br /> | (a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)<br /> | ||
Let c and d be the bezout pair of a+b and b | Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1<br /> | ||
Since the pergen is a double-split, m &gt; 1, therefore |b| &gt; 1, therefore c ≠ 0<br /> | Since the pergen is a double-split, m &gt; 1, therefore |b| &gt; 1, therefore c ≠ 0<br /> | ||
c·(a+b)·P8 = c·b·((n/b)·G - P5)<br /> | c·(a+b)·P8 = c·b·((n/b)·G - P5)<br /> | ||
(1 - d·b)·P8 = c·b·((n/b)·G - P5)<br /> | (1 - d·b)·P8 = c·b·((n/b)·G - P5)<br /> | ||
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)<br /> | P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)<br /> | ||
P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)<br /> | P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)<br /> | ||
Therefore P8 is split into | Therefore P8 is split into m periods<br /> | ||
Therefore if m = |b|, the pergen is explicitly false<br /> | Therefore if m = |b|, the pergen is explicitly false<br /> | ||
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