S-expression: Difference between revisions
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In other words, what this shows is all 1/3-square-particulars of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all 1/3-square-particulars of the form S(3''k'' - 1) * S(3''k'') * S(3''k'' + 1) are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff k is threven and superparticular iff k is throdd). | In other words, what this shows is all 1/3-square-particulars of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all 1/3-square-particulars of the form S(3''k'' - 1) * S(3''k'') * S(3''k'' + 1) are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff k is threven and superparticular iff k is throdd). | ||
Below is a table of such commas in the 41-prime-limited | Below is a table of such commas in the 41-prime-limited 199-odd-limit: | ||
{| class="wikitable center-all | {| class="wikitable center-all | ||
|- | |- | ||
| Line 598: | Line 598: | ||
| ([[96/95]])/([[99/98]]) | | ([[96/95]])/([[99/98]]) | ||
| [[3136/3135]] | | [[3136/3135]] | ||
|- | |||
| S112*S113*S114 | |||
| ([[112/111]])/([[115/114]]) | |||
| [[4256/4255]] | |||
|- | |||
| S117*S118*S119 | |||
| ([[117/116]])/([[120/119]]) | |||
| [[4641/4640]] | |||
|- | |||
| S121*S122*S123 | |||
| ([[121/120]])/([[124/123]]) | |||
| [[4961/4960]] | |||
|- | |||
| S133*S134*S135 | |||
| ([[133/132]])/([[136/135]]) | |||
| [[5985/5984]] | |||
|- | |||
| S145*S146*S147 | |||
| ([[145/144]])/([[148/147]]) | |||
| [[7105/7104]] | |||
|- | |||
| S153*S154*S155 | |||
| ([[153/152]])/([[156/155]]) | |||
| [[7905/7904]] | |||
|- | |||
| S162*S163*S164 | |||
| ([[162/161]])/([[165/164]]) | |||
| [[8856/8855]] | |||
|- | |||
| S187*S188*S189 | |||
| ([[187/186]])/([[190/189]]) | |||
| [[11781/11780]] | |||
|} | |} | ||
Note how there | |||
== S''k''*S(''k'' + 1)*...*S(''k'' + ''n'' - 1) (1/n-square-particulars) == | == S''k''*S(''k'' + 1)*...*S(''k'' + ''n'' - 1) (1/n-square-particulars) == | ||