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== C''k'' and Cp''k'' (cube-particulars) ==
== C''k'' and Cp''k'' (cube-particulars) ==
This family of superparticular interval is of the form {{nowrap|{{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} {{=}} C''k''}} and {{nowrap|{{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}} {{=}} Cp''k''}} (for ''cube-particular complement''). Both C''k'' and Cp''k'' are notable because ''k''<sup>3</sup> + 1 and ''k''<sup>3</sup> − 1 are always composite, unlike with square-particulars, where ''k''<sup>2</sup> + 1 can be prime. The term ''S-expression'' applies to these despite not using the letter ''S'', in avoidance of introducing additional terms.  
This family of superparticular interval is of the form {{nowrap|{{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} {{=}} C''k''}} and {{nowrap|{{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}} {{=}} Cp''k''}} (for ''cube-particular complement''). Both C''k'' and Cp''k'' are notable because ''k''<sup>3</sup> + 1 and ''k''<sup>3</sup> − 1 are always composite for ''k'' ≥ 2, unlike with square-particulars, where ''k''<sup>2</sup> + 1 can be prime. The term ''S-expression'' applies to these despite not using the letter ''S'', in avoidance of introducing additional terms.  


Note that as ''k'' increases, the maximal prime limit of a cube-particular grows more quickly than that of a square-particular; cube-particulars essentially rely on the factorizability of the term (''k''<sup>2</sup> + ''k'' + 1) for C''k'' or (''k''<sup>2</sup> − ''k'' + 1) for Cp''k'' to get to a reasonable prime limit.  
Note that as ''k'' increases, the maximal prime limit of a cube-particular grows more quickly than that of a square-particular; cube-particulars essentially rely on the factorizability of the term (''k''<sup>2</sup> + ''k'' + 1) for C''k'' or (''k''<sup>2</sup> − ''k'' + 1) for Cp''k'' to get to a reasonable prime limit.  
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=== Properties ===
=== Properties ===
# Like square-particulars, all cube-particulars involve prime 2, but unlike square-particulars, some cube-particulars are no-3. These are Cp(3''k'' + 1) and C(3''k'' + 2) for any integer ''k''.
# C''k''/Cp''k'' = C(''k''<sup>2</sup>). Note S''k''/Sp''k'' = S(k<sup>2</sup>) holds too if Sp''k'' notation is used for (''k''<sup>2</sup> + 1)/k<sup>2</sup>.
# C''k''/Cp''k'' = C(''k''<sup>2</sup>). Note S''k''/Sp''k'' = S(k<sup>2</sup>) holds too if Sp''k'' notation is used for (''k''<sup>2</sup> + 1)/k<sup>2</sup>.
# C(''k''<sup>2</sup>) = S(''k''<sup>3</sup>). In other words, C''k'' is a square-particular if ''k'' is a perfect square, and conversely S''k'' is a cube-particular if ''k'' is a perfect cube.
# C(''k''<sup>2</sup>) = S(''k''<sup>3</sup>). In other words, C''k'' is a square-particular if ''k'' is a perfect square, and conversely S''k'' is a cube-particular if ''k'' is a perfect cube.
=== Table of cube-particulars ===
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-all left-3 left-6"
|+ style="font-size: 105%;" | 31-limit cube-particulars – C''k''
|-
! S-expression
! Ratio
! Subgroup
|-
| –
| –
| –
|-
| C2
| [[8/7]]
| 2.7
|-
| C3
| [[27/26]]
| 2.3.13
|-
| C4
| [[64/63]]
| 2.3.7
|-
| C5
| [[125/124]]
| 2.5.31
|-
| C7
| [[343/342]]
| 2.3.7.19
|-
| C9
| [[729/728]]
| 2.3.7.13
|-
| C11
| [[1331/1330]]
| 2.5.7.11.19
|-
| C16
| [[4096/4095]]
| 2.3.5.7.13
|-
| C18
| [[5832/5831]]
| 2.3.7.17
|-
| C22
| [[10648/10647]]
| 2.3.7.11.13
|-
| C25
| [[15625/15624]]
| 2.3.5.7.31
|-
| C30
| [[27000/26999]]
| 2.3.5.7.19.29
|-
| –
| –
| –
|-
| –
| –
| –
|}
</div>
<div style="display: inline-grid;">
{| class="wikitable center-all left-3 left-6"
|+ style="font-size: 105%;" | 31-limit cube-particulars – Cp''k''
|-
! S-expression
! Ratio
! Subgroup
|-
| Cp2
| [[9/8]]
| 2.3
|-
| Cp3
| [[28/27]]
| 2.3.7
|-
| Cp4
| [[65/64]]
| 2.5.13
|-
| Cp5
| [[126/125]]
| 2.3.5.7
|-
| Cp6
| [[217/216]]
| 2.3.7.31
|-
| Cp8
| [[513/512]]
| 2.3.19
|-
| Cp10
| [[1001/1000]]
| 2.5.7.11.13
|-
| Cp12
| [[1729/1728]]
| 2.3.7.13.19
|-
| Cp17
| [[4914/4913]]
| 2.3.7.13.17
|-
| Cp19
| [[6860/6859]]
| 2.5.7.19
|-
| Cp23
| [[12168/12167]]
| 2.3.13.23
|-
| Cp26
| [[17577/17576]]
| 2.3.7.13.31
|-
| Cp31
| [[29792/29791]]
| 2.7.19.31
|-
| Cp68
| <small>[[314433/314432]]</small>
| 2.3.7.17.23.31
|-
| Cp69
| <small>[[328510/328509]]</small>
| 2.3.5.7.13.19.23
|}
</div></div>
Note that C''k'' and Cp(''k'' + 1) tend to share the same highest prime, and as we set the prime limit to 31, each C''k'' almost perfectly matches a Cp(''k'' + 1) – except for Cp68 and Cp69 at the bottom.


== Using S-factorizations to understand the significance of S-expressions ==
== Using S-factorizations to understand the significance of S-expressions ==
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