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S-expression: Difference between revisions

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- alternative S-expressions in the tables as they are off topic for the sections, already covered in #Equivalent S-expressions, and break the flow. Sort the table of equivalent S-expressions from simple to complex. Collapse the tables for 1/4- and 1/5-square-particulars
Eufalesio (talk | contribs)
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821 edits
→Table of square-particulars: Monzo'd and subgroupped the first S-expression table (WIP)
Line 33: Line 33:
! Interval relation
! Interval relation
! Ratio
! Ratio
! Prime limit
! Monzo
! colspan="2" |Prime limit - Subgroup
|-
|-
| S2
| S2
| ([[2/1]])/([[3/2]])
| ([[2/1]])/([[3/2]])
| [[4/3]]
| [[4/3]]
| 3
| {{Monzo|2 -1}}
| colspan="2" |3
|-
|-
| S3
| S3
| ([[3/2]])/([[4/3]])
| ([[3/2]])/([[4/3]])
| [[9/8]]
| [[9/8]]
| 3
| {{Monzo|-3 2}}
| colspan="2" |3
|-
|-
| S4
| S4
| ([[4/3]])/([[5/4]])
| ([[4/3]])/([[5/4]])
| [[16/15]]
| [[16/15]]
| 5
| {{Monzo|4 -1 -1}}
| colspan="2" |5
|-
|-
| S5
| S5
| ([[5/4]])/([[6/5]])
| ([[5/4]])/([[6/5]])
| [[25/24]]
| [[25/24]]
| 5
| {{Monzo|4 -1 -1}}
| colspan="2" |5
|-
|-
| S6
| S6 = S8⋅S9
| ([[6/5]])/([[7/6]])
| ([[6/5]])/([[7/6]])
| [[36/35]]
| [[36/35]]
| 7
| {{Monzo|2 2 -1 -1}}
| colspan="2" |7
|-
|-
| S7
| S7
| ([[7/6]])/([[8/7]])
| ([[7/6]])/([[8/7]])
| [[49/48]]
| [[49/48]]
| 7
| {{Monzo|-4 -1 0 2}}
|7
|2.3.7
|-
|-
| S8
| S8
| ([[8/7]])/([[9/8]])
| ([[8/7]])/([[9/8]])
| [[64/63]]
| [[64/63]]
| 7
| {{Monzo|6 -2 0 -1}}
|7
|2.3.7
|-
|-
| S9
| S9 = S6/S8
| ([[9/8]])/([[10/9]])
| ([[9/8]])/([[10/9]])
| [[81/80]]
| [[81/80]]
| 5
| {{Monzo|-4 4 -1}}
| colspan="2" |5
|-
|-
| S10
| S10
| ([[10/9]])/([[11/10]])
| ([[10/9]])/([[11/10]])
| [[100/99]]
| [[100/99]]
| 11
| {{Monzo|2 -2 2 0 -1}}
|11
|2.3.5.11
|-
|-
| S11
| S11
| ([[11/10]])/([[12/11]])
| ([[11/10]])/([[12/11]])
| [[121/120]]
| [[121/120]]
| 11
| {{Monzo|-3 -1 -1 0 2}}
|11
|2.3.5.11
|-
|-
| S12
| S12
| ([[12/11]])/([[13/12]])
| ([[12/11]])/([[13/12]])
| [[144/143]]
| [[144/143]]
| 13
| {{Monzo|4 2 0 0 -1 -1}}
|13
|2.3.11.13
|-
|-
| S13
| S13
| ([[13/12]])/([[14/13]])
| ([[13/12]])/([[14/13]])
| [[169/168]]
| [[169/168]]
| 13
| {{Monzo|-3 -1 0 -1 0 2}}
|13
|2.3.7.13
|-
|-
| S14
| S14
| ([[14/13]])/([[15/14]])
| ([[14/13]])/([[15/14]])
| [[196/195]]
| [[196/195]]
| 13
| {{Monzo|2 -1 -1 2 0 -1}}
|13
|2.3.5.7.13
|-
|-
| S15
| S15
| ([[15/14]])/([[16/15]])
| ([[15/14]])/([[16/15]])
| [[225/224]]
| [[225/224]]
| 7
| {{Monzo|-5 2 2 -1}}
| colspan="2" |7
|-
|-
| S16
| S16
| ([[16/15]])/([[17/16]])
| ([[16/15]])/([[17/16]])
| [[256/255]]
| [[256/255]]
| 17
| {{Monzo|-8 -1 -1 0 0 0 -1}}
|17
|2.3.5.17
|-
|-
| S17
| S17
| ([[17/16]])/([[18/17]])
| ([[17/16]])/([[18/17]])
| [[289/288]]
| [[289/288]]
| 17
| {{Monzo|5 2 0 0 0 0 -2}}
|17
|2.3.17
|-
|-
| S18
| S18
| ([[18/17]])/([[19/18]])
| ([[18/17]])/([[19/18]])
| [[324/323]]
| [[324/323]]
| 19
| {{Monzo|2 4 0 0 0 0 -1 -1}}
|19
|2.3.17.19
|-
|-
| S19
| S19
| ([[19/18]])/([[20/19]])
| ([[19/18]])/([[20/19]])
| [[361/360]]
| [[361/360]]
| 19
| {{Monzo|-3 -2 -1 0 0 0 0 2}}
|19
|2.3.5.19
|-
|-
| S20
| S20
| ([[20/19]])/([[21/20]])
| ([[20/19]])/([[21/20]])
| [[400/399]]
| [[400/399]]
| 19
| {{Monzo|4 -1 2 -1 0 0 0 -1}}
|19
|2.3.5.7.19
|-
|-
| S21
| S21
| ([[21/20]])/([[22/21]])
| ([[21/20]])/([[22/21]])
| [[441/440]]
| [[441/440]]
| 11
| {{Monzo|-3 2 -1 2 -1}}
| colspan="2" |11
|-
|-
| S22
| S22
| ([[22/21]])/([[23/22]])
| ([[22/21]])/([[23/22]])
| [[484/483]]
| [[484/483]]
| 23
| {{Monzo|2 -1 0 -1 2 0 0 0 -1}}
|23
|2.3.7.11.23
|-
|-
| S23
| S23
| ([[23/22]])/([[24/23]])
| ([[23/22]])/([[24/23]])
| [[529/528]]
| [[529/528]]
| 23
| {{Monzo|-4 -1 0 0 -1 0 0 0 2}}
|23
|2.3.11.23
|-
|-
| S24
| S24
| ([[24/23]])/([[25/24]])
| ([[24/23]])/([[25/24]])
| [[576/575]]
| [[576/575]]
| 23
| {{Monzo|6 2 -2 0 0 0 0 0 -1}}
|23
|2.3.5.23
|-
|-
| S25
| S25
| ([[25/24]])/([[26/25]])
| ([[25/24]])/([[26/25]])
| [[625/624]]
| [[625/624]]
| 13
| {{Monzo|-4 -1 4 0 0 -1}}
|13
|2.3.5.13
|-
|-
| S26
| S26 = S13/S15
| ([[26/25]])/([[27/26]])
| ([[26/25]])/([[27/26]])
| [[676/675]]
| [[676/675]]
| 13
| {{Monzo|2 -3 -2 0 0 2}}
|13
|2.3.5.13
|-
|-
| S27
| S27
| ([[27/26]])/([[28/27]])
| ([[27/26]])/([[28/27]])
| [[729/728]]
| [[729/728]]
| 13
| {{Monzo|-3 6 0 -1 0 -1}}
|13
|2.3.7.13
|-
|-
| S28
| S28
| ([[28/27]])/([[29/28]])
| ([[28/27]])/([[29/28]])
| [[784/783]]
| [[784/783]]
| 29
| {{Monzo|4 -3 0 2 0 0 0 0 0 -1}}
|29
|2.3.7.29
|-
|-
| S29
| S29
| ([[29/28]])/([[30/29]])
| ([[29/28]])/([[30/29]])
| [[841/840]]
| [[841/840]]
| 29
| {{Monzo|-3 -1 -1 -1 0 0 0 0 0 2}}
|29
|2.3.5.7.29
|-
|-
| S30
| S30
| ([[30/29]])/([[31/30]])
| ([[30/29]])/([[31/30]])
| [[900/899]]
| [[900/899]]
| 31
| {{Monzo|2 2 2 0 0 0 0 0 0 -1 -1}}
|31
|2.3.5.29.31
|-
|-
| S31
| S31
| ([[31/30]])/([[32/31]])
| ([[31/30]])/([[32/31]])
| [[961/960]]
| [[961/960]]
| 31
| {{Monzo|-6 -1 -1 0 0 0 0 0 0 0 2}}
|31
|2.3.5.31
|-
|-
| S32
| S32
| ([[32/31]])/([[33/32]])
| ([[32/31]])/([[33/32]])
| [[1024/1023]]
| [[1024/1023]]
| 31
| {{Monzo|10 -1 0 0 -1 0 0 0 0 0 -1}}
|31
|2.3.11.31
|-
|-
| S33
| S33
| ([[33/32]])/([[34/33]])
| ([[33/32]])/([[34/33]])
| [[1089/1088]]
| [[1089/1088]]
| 17
| {{Monzo|-6 2 0 0 2 0 -1}}
|17
|2.3.11.17
|-
|-
| S34
| S34
| ([[34/33]])/([[35/34]])
| ([[34/33]])/([[35/34]])
| [[1156/1155]]
| [[1156/1155]]
| 17
| {{Monzo|2 -1 -1 -1 -1 0 2}}
|17
|2.3.5.7.11.17
|-
|-
| S35
| S35 = S49⋅S50
| ([[35/34]])/([[36/35]])
| ([[35/34]])/([[36/35]])
| [[1225/1224]]
| [[1225/1224]]
| 17
| {{Monzo|-3 -2 2 2 0 0 -1}}
|17
|2.3.5.7.17
|-
|-
| S39
| S39
| ([[39/38]])/([[40/39]])
| ([[39/38]])/([[40/39]])
| [[1521/1520]]
| [[1521/1520]]
| 19
| {{Monzo|-4 2 -1 0 0 2 0 -1}}
|19
|2.3.5.13.19
|-
|-
| S45
| S45
| ([[45/44]])/([[46/45]])
| ([[45/44]])/([[46/45]])
| [[2025/2024]]
| [[2025/2024]]
| 23
| {{Monzo|-3 4 2 0 -1 0 0 0 -1}}
|23
|2.3.5.11.23
|-
|-
| S49
| S49
| ([[49/48]])/([[50/49]])
| ([[49/48]])/([[50/49]])
| [[2401/2400]]
| [[2401/2400]]
| 7
| {{Monzo|-5 -1 -2 4}}
| colspan="2" |7
|-
|-
| S50
| S50
| ([[50/49]])/([[51/50]])
| ([[50/49]])/([[51/50]])
| [[2500/2499]]
| [[2500/2499]]
| 17
| {{Monzo|2 -1 4 -2 0 0 -1}}
|17
|2.3.5.7.17
|-
|-
| S51
| S51
| ([[51/50]])/([[52/51]])
| ([[51/50]])/([[52/51]])
| [[2601/2600]]
| [[2601/2600]]
| 17
| {{Monzo|-3 2 -2 0 0 -1 2}}
|17
|2.3.5.13.17
|-
|-
| S55
| S55 = S22/S24
| ([[55/54]])/([[56/55]])
| ([[55/54]])/([[56/55]])
| [[3025/3024]]
| [[3025/3024]]
| 11
| {{Monzo|-4 -3 2 -1 2}}
| colspan="2" |11
|-
|-
| S56
| S56
| ([[56/55]])/([[57/56]])
| ([[56/55]])/([[57/56]])
| [[3136/3135]]
| [[3136/3135]]
| 19
| {{Monzo|6 -1 -1 2 -1 0 0 -1}}
|19
|2.3.5.7.11.19
|-
|-
| S57
| S57
| ([[57/56]])/([[58/57]])
| ([[57/56]])/([[58/57]])
| [[3249/3248]]
| [[3249/3248]]
| 29
| {{Monzo|-4 2 0 -1 0 0 0 2 0 -1}}
|29
|2.3.7.19.29
|-
|-
| S63
| S63
| ([[63/62]])/([[64/63]])
| ([[63/62]])/([[64/63]])
| [[3969/3968]]
| [[3969/3968]]
| 31
| {{Monzo|-7 4 0 2 0 0 0 0 0 0 -1}}
|31
|2.3.7.31
|-
|-
| S64
| S64
| ([[64/63]])/([[65/64]])
| ([[64/63]])/([[65/64]])
| [[4096/4095]]
| [[4096/4095]]
| 13
| {{Monzo|12 -2 -1 -1 0 -1}}
|13
|2.3.5.7.13
|-
|-
| S65
| S65
| ([[65/64]])/([[66/65]])
| ([[65/64]])/([[66/65]])
| [[4225/4224]]
| [[4225/4224]]
| 13
| {{Monzo|-7 -1 2 0 -1 2}}
|13
|2.3.5.11.13
|-
|-
| S69
| S69
| ([[69/68]])/([[70/69]])
| ([[69/68]])/([[70/69]])
| [[4761/4760]]
| [[4761/4760]]
| 23
| {{Monzo|-3 2 -1 -1 0 0 -1 0 2}}
|23
|2.3.5.7.17.23
|-
|-
| S76
| S76
| ([[76/75]])/([[77/76]])
| ([[76/75]])/([[77/76]])
| [[5776/5775]]
| [[5776/5775]]
| 19
| {{Monzo|4 -1 -2 -1 -1 0 0 2}}
|19
|2.3.5.7.11.19
|-
|-
| S77
| S77
| ([[77/76]])/([[78/77]])
| ([[77/76]])/([[78/77]])
| [[5929/5928]]
| [[5929/5928]]
| 19
| {{Monzo|-3 -1 0 2 2 -1 0 -1}}
|19
|2.3.7.11.13.19
|-
|-
| S91
| S91
| ([[91/90]])/([[92/91]])
| ([[91/90]])/([[92/91]])
| [[8281/8280]]
| [[8281/8280]]
| 23
| {{Monzo|-3 -2 -1 2 0 2 0 0 -1}}
|23
|2.3.5.7.13.23
|-
|-
| S92
| S92
| ([[92/91]])/([[93/92]])
| ([[92/91]])/([[93/92]])
| [[8464/8463]]
| [[8464/8463]]
| 31
| {{Monzo|4 -1 0 -1 0 -1 0 0 2 0 -1}}
|31
|2.3.7.13.23.31
|-
|-
| S99
| S99 = S33/S35
| ([[99/98]])/([[100/99]])
| ([[99/98]])/([[100/99]])
| [[9801/9800]]
| [[9801/9800]]
| 11
| {{Monzo|-3 4 -2 -2 2}}
| colspan="2" |11
|-
|-
| S115
| S115
| ([[115/114]])/([[116/115]])
| ([[115/114]])/([[116/115]])
| [[13225/13224]]
| [[13225/13224]]
| 29
|
|29
|
|-
|-
| S116
| S116
| ([[116/115]])/([[117/116]])
| ([[116/115]])/([[117/116]])
| [[13456/13455]]
| [[13456/13455]]
| 29
|
|29
|
|-
|-
| S120
| S120
| ([[120/119]])/([[121/120]])
| ([[120/119]])/([[121/120]])
| [[14400/14399]]
| [[14400/14399]]
| 17
| {{Monzo|6 2 2 -1 -2 0 -1}}
|17
|2.3.5.7.11.17
|-
|-
| S125
| S125
| ([[125/124]])/([[126/125]])
| ([[125/124]])/([[126/125]])
| [[15625/15624]]
| [[15625/15624]]
| 31
|
|31
|
|-
|-
| S144
| S144
| ([[144/143]])/([[145/144]])
| ([[144/143]])/([[145/144]])
| [[20736/20735]]
| [[20736/20735]]
| 29
|
|29
|
|-
|-
| S153
| S153
| ([[153/152]])/([[154/153]])
| ([[153/152]])/([[154/153]])
| [[23409/23408]]
| [[23409/23408]]
| 19
|
|19
|
|-
|-
| S154
| S154
| ([[154/153]])/([[155/154]])
| ([[154/153]])/([[155/154]])
| [[23716/23715]]
| [[23716/23715]]
| 31
|
|31
|
|-
|-
| S155
| S155
| ([[155/154]])/([[156/155]])
| ([[155/154]])/([[156/155]])
| [[24025/24024]]
| [[24025/24024]]
| 31
|
|31
|
|-
|-
| S161
| S161 = S46/S48
| ([[161/160]])/([[162/161]])
| ([[161/160]])/([[162/161]])
| [[25921/25920]]
| [[25921/25920]]
| 23
| {{Monzo|-6 -4 -1 2 0 0 0 0 2}}
|23
|2.3.5.7.23
|-
|-
| S169
| S169
| ([[169/168]])/([[170/169]])
| ([[169/168]])/([[170/169]])
| [[28561/28560]]
| [[28561/28560]]
| 17
| {{Monzo|-4 -1 -1 -1 0 4 -1}}
|17
|2.3.5.7.13.17
|-
|-
| S170
| S170
| ([[170/169]])/([[171/170]])
| ([[170/169]])/([[171/170]])
| [[28900/28899]]
| [[28900/28899]]
| 19
|
|19
|
|-
|-
| S175
| S175
| ([[175/174]])/([[176/175]])
| ([[175/174]])/([[176/175]])
| [[30625/30624]]
| [[30625/30624]]
| 29
|
|29
|
|-
|-
| S208
| S208
| ([[208/207]])/([[209/208]])
| ([[208/207]])/([[209/208]])
| [[43264/43263]]
| [[43264/43263]]
| 23
|
|23
|
|-
|-
| S209
| S209
| ([[209/208]])/([[210/209]])
| ([[209/208]])/([[210/209]])
| [[43681/43680]]
| [[43681/43680]]
| 19
|
|19
|
|-
|-
| S231
| S231
| ([[231/230]])/([[232/231]])
| ([[231/230]])/([[232/231]])
| [[53361/53360]]
| [[53361/53360]]
| 29
|
|29
|
|-
|-
| S289
| S289
| ([[289/288]])/([[290/289]])
| ([[289/288]])/([[290/289]])
| [[83521/83520]]
| [[83521/83520]]
| 29
|
|29
|
|-
|-
| S323
| S323
| ([[323/322]])/([[324/323]])
| ([[323/322]])/([[324/323]])
| [[104329/104328]]
| [[104329/104328]]
| 23
|
|23
|
|-
|-
| S324
| S324
| ([[324/323]])/([[325/324]])
| ([[324/323]])/([[325/324]])
| [[104976/104975]]
| [[104976/104975]]
| 19
|
|19
|
|-
|-
| S341
| S341
| ([[341/340]])/([[342/341]])
| ([[341/340]])/([[342/341]])
| [[116281/116280]]
| [[116281/116280]]
| 31
|
|31
|
|-
|-
| S342
| S342
| ([[342/341]])/([[343/342]])
| ([[342/341]])/([[343/342]])
| [[116964/116963]]
| [[116964/116963]]
| 31
|
|31
|
|-
|-
| S351
| S351 = S78/S80
| ([[351/350]])/([[352/351]])
| ([[351/350]])/([[352/351]])
| [[123201/123200]]
| [[123201/123200]]
| 13
| {{Monzo|-6 6 -2 -1 -1 2}}
| colspan="2" |13
|-
|-
| S391
| S391
| ([[391/390]])/([[392/391]])
| ([[391/390]])/([[392/391]])
| [[152881/152880]]
| [[152881/152880]]
| 23
|
|23
|
|-
|-
| S441
| S441
| ([[441/440]])/([[442/441]])
| ([[441/440]])/([[442/441]])
| [[194481/194480]]
| [[194481/194480]]
| 17
| {{Monzo|-4 4 -1 4 -1 -1 -1}}
| colspan="2" |17
|-
|-
| S494
| S494
| ([[494/493]])/([[495/494]])
| ([[494/493]])/([[495/494]])
| [[244036/244035]]
| [[244036/244035]]
| 29
|
|29
|
|-
|-
| S495
| S495
| ([[495/494]])/([[496/495]])
| ([[495/494]])/([[496/495]])
| [[245025/245024]]
| [[245025/245024]]
| 31
|
|31
|
|-
|-
| S528
| S528
| ([[528/527]])/([[529/528]])
| ([[528/527]])/([[529/528]])
| [[278784/278783]]
| [[278784/278783]]
| 31
|
|31
|
|-
|-
| S551
| S551
| ([[551/550]])/([[552/551]])
| ([[551/550]])/([[552/551]])
| [[303601/303600]]
| [[303601/303600]]
| 29
|
|29
|
|-
|-
| S714
| S714
| ([[714/713]])/([[715/714]])
| ([[714/713]])/([[715/714]])
| [[509796/509795]]
| [[509796/509795]]
| 31
|
|31
|
|-
|-
| S783
| S783
| ([[783/782]])/([[784/783]])
| ([[783/782]])/([[784/783]])
| [[613089/613088]]
| [[613089/613088]]
| 29
|
|29
|
|-
|-
| S1275
| S1275
| ([[1275/1274]])/([[1276/1275]])
| <small>([[1275/1274]])/([[1276/1275]])</small>
| [[1625625/1625624]]
| <small>[[1625625/1625624]]</small>
| 29
|
|29
|
|-
|-
| S1519
| S1519
| ([[1519/1518]])/([[1520/1519]])
| <small>([[1519/1518]])/([[1520/1519]])</small>
| [[2307361/2307360]]
| <small>[[2307361/2307360]]</small>
| 31
|
|31
|
|-
|-
| S1520
| S1520
| ([[1520/1519]])/([[1521/1520]])
| <small>([[1520/1519]])/([[1521/1520]])</small>
| [[2310400/2310399]]
| <small>[[2310400/2310399]]</small>
| 31
|
|31
|
|-
|-
| S2001
| S2001
| ([[2001/2000]])/([[2002/2001]])
| <small>([[2001/2000]])/([[2002/2001]])</small>
| [[4004001/4004000]]
| <small>[[4004001/4004000]]</small>
| 29
| {{Monzo|-5 2 -3 -1 -1 -1 0 0 2 2}}
|29
|2.3.5.7.11.13.23.29
|-
|-
| S2024
| S2024
| ([[2024/2023]])/([[2025/2024]])
| <small>([[2024/2023]])/([[2025/2024]])</small>
| [[4096576/4096575]]
| <small>[[4096576/4096575]]</small>
| 23
|
|23
|
|-
|-
| S2431
| S2431
| ([[2431/2430]])/([[2432/2431]])
| <small>([[2431/2430]])/([[2432/2431]])</small>
| [[5909761/5909760]]
| <small>[[5909761/5909760]]</small>
| 19
|
|19
|
|-
|-
| S3249
| S3249
| ([[3249/3248]])/([[3250/3249]])
| <small>([[3249/3248]])/([[3250/3249]])</small>
| <font style="font-size:0.88em">[[10556001/10556000]]</font>
| <small>[[10556001/10556000]]</small>
| 29
|
|29
|
|-
|-
| S9801
| S9801
| ([[9801/9800]])/([[9802/9801]])
| <small>([[9801/9800]])/([[9802/9801]])</small>
| <font style="font-size:0.88em">[[96059601/96059600]]</font>
| <small>[[96059601/96059600]]</small>
| 29
|
|29
|
|-
|-
| S13311
| S13311
| <font style="font-size:0.86em">([[13311/13310]])/([[13312/13311]])</font>
| <small><small>([[13311/13310]])/([[13312/13311]])</small></small>
| <font style="font-size:0.79em">[[177182721/177182720]]</font>
| <small><small>[[177182721/177182720]]</small></small>
| 29
|
|29
|
|-
|-
| S13455
| S13455
| <font style="font-size:0.86em">([[13455/13454]])/([[13456/13455]])</font>
| <small><small>([[13455/13454]])/([[13456/13455]])</small></small>
| <font style="font-size:0.79em">[[181037025/181037024]]</font>
| <small><small>[[181037025/181037024]]</small></small>
| 31
|
|31
|
|}
|}


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