S-expression: Difference between revisions

Godtone (talk | contribs)
Godtone (talk | contribs)
Line 384: Line 384:
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 99-odd-limit:
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) ==