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Squib

Joined 25 April 2025
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(For subgroups with rational or non-prime elements, split them into prime factors and multiply all together to get n, then determine if the final result is in the subgroup. For example, for the 11/2.13.15.19 subgroup, n is 81510 and (4n+1)^0.5 is 571, so the resulting superparticular interval is 286/285, but this is not in the subgroup because 11 and 2 are on the same side of the fraction. So no superparticular interval exists in the subgroup.)
(For subgroups with rational or non-prime elements, split them into prime factors and multiply all together to get n, then determine if the final result is in the subgroup. For example, for the 11/2.13.15.19 subgroup, n is 81510 and (4n+1)^0.5 is 571, so the resulting superparticular interval is 286/285, but this is not in the subgroup because 11 and 2 are on the same side of the fraction. So no superparticular interval exists in the subgroup.)


(note about commas like 245/243)
(note about intervals like 35/33)


(this should probably get its own page lol)
(this should probably get its own page lol)


====All superparticular intervals with no duplicate primes, by prime limit====
====All superparticular intervals with no duplicate primes, by prime limit====
Found by applying this method to every possible subgroup in the prime limit, using [https://www.desmos.com/calculator/2o1mwkkxf9 this desmos graph].
Found by applying this method to every possible subgroup in the prime limit, using [https://www.desmos.com/calculator/0qnrxfzey0 this desmos graph].
 
{| class="wikitable"
2-limit:
|+
*[[2/1]]
|
 
!1 (superparticular)
3-limit:
!2
*[[3/2]]
!3
 
|-
5-limit:
!2-limit
*[[6/5]]
|[[2/1]]
 
| -
7-limit:
| -
*[[7/6]]
|-
*[[15/14]]
!3-limit
 
|[[3/2]]
11-limit:
|[[3/1]]
*[[11/10]]
| -
*[[22/21]]
|-
 
!5-limit
13-limit:
|[[6/5]]
*[[14/13]]
|[[5/3]]
*[[66/65]]
|[[5/2]]
*[[78/77]]
|-
 
!7-limit
17-limit:
|[[7/6]], [[15/14]]
*[[34/33]]
|[[7/5]]
*[[35/34]]
|[[10/7]]
*[[715/714]]
|-
 
!11-limit
19-limit:
|[[11/10]], [[22/21]]
*[[39/38]]
|[[35/33]]
*[[210/209]]
|[[14/11]]
*[[286/285]]
|-
 
!13-limit
23-limit:
|[[14/13]], [[66/65]], [[78/77]]
*[[23/22]]
|[[13/11]], [[15/13]]
*[[70/69]]
|[[13/10]]
*[[115/114]]
|-
*[[231/230]]
!17-limit
*[[323/322]]
|[[34/33]], [[35/34]], [[715/714]]
*[[391/390]]
|[[17/15]]
 
|[[17/14]]
29-limit:
|-
*[[30/29]]
!19-limit
*[[58/57]]
|[[39/38]], [[210/209]], [[286/285]]
*[[494/493]]
|[[19/17]], [[21/19]], [[57/55]], [[665/663]]
*[[2002/2001]]
|[[22/19]], [[38/35]], [[133/130]], [[190/187]]
*[[2262/2261]]
|-
 
!23-limit
31-limit:
|[[23/22]], [[70/69]], [[115/114]], [[231/230]], [[323/322]], [[391/390]]
*[[31/30]]
|[[23/21]], [[255/253]], [[1311/1309]]
*[[155/154]]
|[[26/23]], [[598/595]], [[2093/2090]]
*[[187/186]]
|-
*[[435/434]]
!29-limit
*[[714/713]]
|[[30/29]], [[58/57]], [[494/493]], [[2002/2001]], [[2262/2261]]
*[[806/805]]
|[[87/85]], [[145/143]], [[437/435]], [[667/665]]
*[[12122/12121]]
|[[29/26]], [[58/55]], [[322/319]], [[377/374]], [[1105/1102]]
|-
!31-limit
|[[31/30]], [[155/154]], [[187/186]], [[435/434]], [[714/713]], [[806/805]], [[12122/12121]]
|[[31/29]], [[33/31]], [[93/91]], [[95/93]], [[715/713]], [[899/897]], [[7163/7161]]
|[[34/31]], [[65/62]], [[406/403]], [[437/434]], [[10013/10010]]
|}
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