Delta-N ratio: Difference between revisions

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<math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>

Likewise, you can also express any delta-''N'' ratio as a product of any number of delta-''M'' ratios with ''M'' being a divisor of ''N''. For example,

* 5/3 = (10/8) (8/6) = '''(5/4) (4/3)''' ''— can’t get delta-2 because we have an even number of factors.''

* 5/3 = (15/13) (13/11) (11/9).

* 5/3 = (20/18) (18/16) (16/14) (14/12) = '''(10/9) (9/8) (8/7) (7/6)''' ''— again only delta-1.''

* 5/3 = (25/23) (23/21) (21/19) (19/17) (17/15).

* 4/1 = (8/5) (5/2).

* 4/1 = (12/9) (9/6) (6/3) = '''(4/3) (3/2) (2/1)''' ''— now we can’t get delta-3 because there are 3 factors.''

* 4/1 = (16/13) (13/10) (10/7) (7/4).

Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.

[[Wikipedia:Størmer's theorem|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

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 <math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>
+Likewise, you can also express any delta-''N'' ratio as a product of any number of delta-''M'' ratios with ''M'' being a divisor of ''N''. For example,
+* 5/3 = (10/8) (8/6) = '''(5/4) (4/3)''' ''— can’t get delta-2 because we have an even number of factors.''
+* 5/3 = (15/13) (13/11) (11/9).
+* 5/3 = (20/18) (18/16) (16/14) (14/12) = '''(10/9) (9/8) (8/7) (7/6)''' ''— again only delta-1.''
+* 5/3 = (25/23) (23/21) (21/19) (19/17) (17/15).
+* 4/1 = (8/5) (5/2).
+* 4/1 = (12/9) (9/6) (6/3) = '''(4/3) (3/2) (2/1)''' ''— now we can’t get delta-3 because there are 3 factors.''
+* 4/1 = (16/13) (13/10) (10/7) (7/4).
+Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.
 [[Wikipedia:Størmer's theorem|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.