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List of superparticular intervals: Difference between revisions

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This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 127-limit and is complete up to the [[29-limit]].
This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 127-limit and is complete up to the [[31-limit]].


[[Wikipedia: Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than [[2/1]], [[3/2]], [[4/3]], and [[9/8]]. {{OEIS| A002071 }} gives the number of superparticular ratios in each prime limit, {{OEIS| A145604 }} shows the increment from limit to limit, and {{OEIS| A117581 }} gives the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).
[[Wikipedia: Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than [[2/1]], [[3/2]], [[4/3]], and [[9/8]]. {{OEIS| A002071 }} gives the number of superparticular ratios in each prime limit, {{OEIS| A145604 }} shows the increment from limit to limit, and {{OEIS| A117581 }} gives the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).
Line 1,246: Line 1,246:
| S2431
| S2431
|-
|-
| 11859211/11859210
| <font style="font-size:0.88em">11859211/11859210</font>
| 0.00014598
| 0.00014598
| (19/(3×11))<sup>4</sup>×((7×13)/(2×5))
| (19/(3×11))<sup>4</sup>×((7×13)/(2×5))
Line 2,491: Line 2,491:
|  
|  
|-
|-
| 10556001/10556000
| <font style="font-size:0.88em">10556001/10556000</font>
| 0.00016400
| 0.00016400
| (3×19)<sup>4</sup>/(2<sup>5</sup>×5<sup>3</sup>×7×13×29)
| (3×19)<sup>4</sup>/(2<sup>5</sup>×5<sup>3</sup>×7×13×29)
Line 2,498: Line 2,498:
| S3249
| S3249
|-
|-
| 18085705/18085704
| <font style="font-size:0.88em">18085705/18085704</font>
| 9.5724×10<sup>-5</sup>
| 9.5724×10<sup>-5</sup>
| (5×11×17×23×29<sup>2</sup>)/((2×7×13)<sup>3</sup>×3)
| (5×11×17×23×29<sup>2</sup>)/((2×7×13)<sup>3</sup>×3)
Line 2,505: Line 2,505:
|  
|  
|-
|-
| 96059601/96059600
| <font style="font-size:0.88em">96059601/96059600</font>
| 1.8022×10<sup>-5</sup>
| 1.8022×10<sup>-5</sup>
| (3<sup>2</sup>×11)<sup>4</sup>/((2<sup>2</sup>×5×7×13)<sup>2</sup>×29)
| (3<sup>2</sup>×11)<sup>4</sup>/((2<sup>2</sup>×5×7×13)<sup>2</sup>×29)
Line 2,512: Line 2,512:
| S9801
| S9801
|-
|-
| 177182721/177182720
| <font style="font-size:0.79em">177182721/177182720</font>
| 9.7709×10<sup>-6</sup>
| 9.7709×10<sup>-6</sup>
| (3<sup>3</sup>×17×29)<sup>2</sup>/(2<sup>11</sup>×5×11<sup>3</sup>×13)
| (3<sup>3</sup>×17×29)<sup>2</sup>/(2<sup>11</sup>×5×11<sup>3</sup>×13)
Line 2,520: Line 2,520:
|}
|}


=== 31-limit (incomplete) ===
=== 31-limit ===
{| class="wikitable center-6" style="width:100%"
{| class="wikitable center-6" style="width:100%"
! width="10%" | [[Ratio]]
! width="10%" | [[Ratio]]
Line 3,380: Line 3,380:
| (2<sup>6</sup>×3×19<sup>2</sup>×23)/((5×11)<sup>2</sup>×17×31)
| (2<sup>6</sup>×3×19<sup>2</sup>×23)/((5×11)<sup>2</sup>×17×31)
| 2.3.5.11.17.19.23.31 {{monzo| 6 1 -2 -2 -1 2 1 -1 }}
| 2.3.5.11.17.19.23.31 {{monzo| 6 1 -2 -2 -1 2 1 -1 }}
|
|
|-
| 2307361/2307360
| 0.00075031
| (7<sup>2</sup>×31)<sup>2</sup>/(2<sup>5</sup>×3×5×11×19×23)
| 2.3.5.7.11.19.23.31 {{monzo| -5 -1 -1 4 -1 -1 -1 2 }}
|
| S1519
|-
| 2310400/2310399
| 0.00074932
| ((2<sup>4</sup>×5×19)/(3×7×13))<sup>2</sup>/31
| 2.3.5.7.13.19.31 {{monzo| 8 -2 2 -2 -2 2 -1 }}
|
| S1520
|-
| 2345057/2345056
| 0.00073825
| (11×13×23<sup>2</sup>×31)/(2<sup>5</sup>×7×19<sup>2</sup>×29)
| 2.7.11.13.19.23.29.31 {{monzo| -5 -1 1 1 -2 2 -1 1 }}
|
|
|-
| 3206269/3206268
| 0.00053995
| (11×19×23<sup>2</sup>×29)/((2×3×13)<sup>2</sup>×17×31)
| 2.3.11.13.17.19.23.29.31 {{monzo| -2 -2 1 -2 -1 1 2 1 -1 }}
|
|
|-
| 3301376/3301375
| 0.00052440
| (2<sup>13</sup>×13×31)/(5<sup>3</sup>×7<sup>4</sup>×11)
| 2.5.7.11.13.31 {{monzo| 13 -3 -4 -1 1 1 }}
|
|
|-
| 3346110/3346109
| 0.00051739
| (2×3<sup>9</sup>×5×17)/(13×19<sup>2</sup>×23×31)
| 2.3.5.13.17.19.23.31 {{monzo| 1 9 1 -1 1 -2 -1 -1 }}
|
|
|-
| 3897166/3897165
| 0.00044423
| (2×7<sup>3</sup>×13×19×23)/(3×5×17<sup>2</sup>×29×31)
| 2.3.5.7.13.17.19.23.29.31 {{monzo| 1 -1 -1 3 1 -2 1 1 -1 -1 }}
|
|
|-
| <font style="font-size:0.88em">14753025/14753024</font>
| 0.00011735
| ((3×5)<sup>2</sup>×7×17×19×29)/(2<sup>8</sup>×11×13<sup>2</sup>×31)
| 2.3.5.7.11.13.17.19.29.31 {{monzo| -8 2 2 1 -1 -2 1 1 1 -1 }}
|
|
|-
| <font style="font-size:0.88em">16093000/16092999</font>
| 0.00010758
| ((2×5)<sup>3</sup>×7×11<sup>2</sup>×19)/(3<sup>4</sup>×13×17×29×31)
| 2.3.5.7.11.13.17.19.29.31 {{monzo| 3 -4 3 1 2 -1 -1 1 -1 -1 }}
|
|
|-
| <font style="font-size:0.88em">76271625/76271624</font>
| 2.2698×10<sup>-5</sup>
| (3<sup>9</sup>×5<sup>3</sup>×31)/((2×11)<sup>3</sup>×13×19×29)
| 2.3.5.11.13.19.29.31 {{monzo| -3 9 3 -3 -1 -1 -1 1 }}
|
|
|-
| <font style="font-size:0.88em">80061345/80061344</font>
| 2.1624×10<sup>-5</sup>
| (3<sup>3</sup>×5×7<sup>4</sup>×13×19)/(2<sup>5</sup>×11<sup>2</sup>×23×29×31)
| 2.3.5.7.11.13.19.23.29.31 {{monzo| -5 3 1 4 -2 1 1 -1 -1 -1 }}
|
|
|-
| <font style="font-size:0.79em">133920000/133919999</font>
| 1.2927×10<sup>-5</sup>
| (2<sup>8</sup>×3<sup>3</sup>×5<sup>4</sup>×31)/(17<sup>2</sup>×19×29<sup>3</sup>)
| 2.3.5.17.19.29.31 {{monzo| 8 3 4 -2 -1 -3 1 }}
|
|
|-
| <font style="font-size:0.79em">181037025/181037024</font>
| 9.5629×10<sup>-6</sup>
| (3<sup>2</sup>×5×13×23)<sup>2</sup>/(2<sup>5</sup>×7×(29×31)<sup>2</sup>)
| 2.3.5.7.13.23.29.31 {{monzo| -5 4 2 -1 2 2 -2 -2 }}
|
| S13455
|-
| <font style="font-size:0.79em">370256250/370256249</font>
| 4.6758×10<sup>-6</sup>
| (2×3×5<sup>5</sup>×7<sup>2</sup>×13×31)/(11<sup>7</sup>×19)
| 2.3.5.7.11.13.19.31 {{monzo| 1 1 5 2 -7 1 -1 1 }}
|
|
|-
| <font style="font-size:0.72em">1611308700/1611308699</font>
| 1.0744×10<sup>-6</sup>
| ((2×3<sup>3</sup>×5×31)/(7<sup>2</sup>×13×19))<sup>2</sup>×(23/11)
| 2.3.5.7.11.13.19.23.31 {{monzo| 2 6 2 -4 -1 -2 -2 1 2 }}
|  
|  
|  
|  
Line 5,181: Line 5,286:
| S71
| S71
|-
|-
| [[Borcherdsma|160561400000 / 160561399999]]
| <font style="font-size:0.60em">[[Borcherdsma|160561400000/160561399999]]</font>
| 1.0783×10<sup>-8</sup>
| 1.0783×10<sup>-8</sup>
| (2<sup>6</sup>×5<sup>5</sup>×19×29×31×47) / (7×11<sup>2</sup>×13×59<sup>3</sup>×71)
| (2<sup>6</sup>×5<sup>5</sup>×19×29×31×47) / (7×11<sup>2</sup>×13×59<sup>3</sup>×71)
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