User:Contribution/Successive superparticular complementary pair: Difference between revisions

For each pair of superparticular ratios <math>{s1}/{s2}</math>​ and <math>{s2}/{s3}</math>, there exists a ratio <math>{a}/{b}</math> such that <math>{s1}/{s2}</math>​ and <math>{s2}/{s3}</math>​ are <math>{a}/{b}</math> complementary; it is observed that <math>a−b=1</math> or <math>a−b=2</math>.

In other words, for each ratio <math>a/b</math> where <math>a−b=1</math> or <math>a−b=2</math>, there exists a pair of superparticular ratios <math>{s1}/{s2}</math>​ and <math>{s2}/{s3}</math> that are <math>{a}/{b}</math> complementary.

We can observe a converging sequence and pattern for low errors: 5, 7, 12; then 7, 9, 16; then 9, 11, 20; then 11, 13, 24; then 13, 15, 28; then 15, 17, 32; then 17, 19, 36; then 19, 21, 40; then 21, 23, 44; etc. --

{{todo|Table|inline=1|comment=Show the table for 3/1, 2/1, 5/3, 3/2, 7/5, 4/3, 9/7, 5/4, 11/9, 6/5. Highlight the pattern in bold.}}

{{todo|Finish the article and move|inline=1|comment=When the article is finished, move it to the main root}}

! #ed3/1 !! 2/1, 3/2 error !! #ed2/1 !! 3/2, 4/3 error !! #ed5/3 !! 4/3, 5/4 error !! #ed3/2 !! 5/4, 6/5 error !! #ed7/5 !! 6/5, 7/6 error !! #ed4/3 !! 7/6, 8/7 error !! #ed9/7 !! 8/7, 9/8 error !! #ed5/4 !! 9/8, 10/9 error !! #ed11/9 !! 10/9, 11/10 error !! #ed6/5 !! 11/10, 12/11 error

| '''[[3edt|<u>3ed3/1</u>]]''' ||'''<u>67.97</u>''' ||[[3edo|3ed2/1]] || 98.04 || [[3ed5/3]] || 91.53 || [[3edf|3ed3/2]] || 81.66 || [[3ed7/5]] || 72.70 || [[3ed4/3]] || 65.16 || [[3ed9/7]] || 58.88 || [[3ed5/4]] || 53.63 || [[3ed11/9]] || 49.20 || [[3ed6/5]] || 45.42

| '''[[5edt|5ed3/1]]''' ||'''58.83''' ||'''[[5edo|5ed2/1]]''' ||'''18.04''' ||[[5ed5/3]] || 32.57 || [[5edf|5ed3/2]] || 34.86 || [[5ed7/5]] || 33.87 || [[5ed4/3]] || 31.96 || [[5ed9/7]] || 29.88 || [[5ed5/4]] || 27.88 || [[5ed11/9]] || 26.04 || [[5ed6/5]] || 24.38

| [[7edt|7ed3/1]] || 113.17 || '''[[7edo|7ed2/1]]''' ||'''16.24''' ||'''[[7ed5/3]]''' ||'''7.30''' ||[[7edf|7ed3/2]] || 14.80 || [[7ed7/5]] || 17.22 || [[7ed4/3]] || 17.73 || [[7ed9/7]] || 17.45 || [[7ed5/4]] || 16.84 || [[7ed11/9]] || 16.12 || [[7ed6/5]] || 15.36

| '''[[8edt|8ed3/1]]''' ||'''11.28''' ||[[8edo|8ed2/1]] || 48.04 || [[8ed5/3]] || 54.68 || [[8edf|8ed3/2]] || 35.34 || [[8ed7/5]] || 24.39 || [[8ed4/3]] || 17.85 || [[8ed9/7]] || 13.63 || [[8ed5/4]] || 10.75 || [[8ed11/9]] || 8.70 || [[8ed6/5]] || 7.18

| [[9edt|9ed3/1]] || 67.97 || [[9edo|9ed2/1]] || 35.29 || '''[[9ed5/3]]''' ||'''6.73''' ||'''[[9edf|9ed3/2]]''' ||'''3.66''' ||[[9ed7/5]] || 7.98 || [[9ed4/3]] || 9.82 || [[9ed9/7]] || 10.54 || [[9ed5/4]] || 10.71 || [[9ed11/9]] || 10.60 || [[9ed6/5]] || 10.35

| [[11edt|11ed3/1]] || 10.34 || [[11edo|11ed2/1]] || 47.41 || [[11ed5/3]] || 15.67 || '''[[11edf|11ed3/2]]''' ||'''3.43''' ||'''[[11ed7/5]]''' ||'''2.09''' ||[[11ed4/3]] || 4.79 || [[11ed9/7]] || 6.14 || [[11ed5/4]] || 6.81 || [[11ed11/9]] || 7.09 || [[11ed6/5]] || 7.16

| [[12edt|12ed3/1]] || 67.97 || '''[[12edo|12ed2/1]]''' ||'''1.96''' ||[[12ed5/3]] || 17.83 || [[12edf|12ed3/2]] || 23.16 || [[12ed7/5]] || 24.16 || [[12ed4/3]] || 17.85 || [[12ed9/7]] || 13.63 || [[12ed5/4]] || 10.75 || [[12ed11/9]] || 8.70 || [[12ed6/5]] || 7.18

| [[13edt|13ed3/1]] || 29.57 || [[13edo|13ed2/1]] || 36.51 || [[13ed5/3]] || 21.85 || [[13edf|13ed3/2]] || 8.34 || '''[[13ed7/5]]''' ||'''1.98''' ||'''[[13ed4/3]]''' ||'''1.31''' ||[[13ed9/7]] || 3.10 || [[13ed5/4]] || 4.11 || [[13ed11/9]] || 4.66 || [[13ed6/5]] || 4.96

| [[15edt|15ed3/1]] || 58.83 || [[15edo|15ed2/1]] || 18.04 || [[15ed5/3]] || 26.39 || [[15edf|15ed3/2]] || 11.94 || [[15ed7/5]] || 4.97 || '''[[15ed4/3]]''' ||'''1.25''' ||'''[[15ed9/7]]''' ||'''0.87''' ||[[15ed5/4]] || 2.12 || [[15ed11/9]] || 2.88 || [[15ed6/5]] || 3.34

| [[16edt|16ed3/1]] || 11.28 || [[16edo|16ed2/1]] || 26.96 || '''[[16ed5/3]]''' ||'''0.59''' ||[[16edf|16ed3/2]] || 8.54 || [[16ed7/5]] || 12.02 || [[16ed4/3]] || 13.28 || [[16ed9/7]] || 13.56 || [[16ed5/4]] || 10.75 || [[16ed11/9]] || 8.70 || [[16ed6/5]] || 7.18

| [[17edt|17ed3/1]] || 30.68 || [[17edo|17ed2/1]] || 3.93 || [[17ed5/3]] || 22.17 || [[17edf|17ed3/2]] || 14.69 || [[17ed7/5]] || 7.25 || [[17ed4/3]] || 3.20 || '''[[17ed9/7]]''' ||'''0.84''' ||'''[[17ed5/4]]''' ||'''0.61''' ||[[17ed11/9]] || 1.52 || [[17ed6/5]] || 2.10

| [[19edt|19ed3/1]] || 1.23 || [[19edo|19ed2/1]] || 7.22 || [[19ed5/3]] || 13.95 || [[19edf|19ed3/2]] || 16.86 || [[19ed7/5]] || 9.06 || [[19ed4/3]] || 4.74 || [[19ed9/7]] || 2.18 || '''[[19ed5/4]]''' ||'''0.59''' ||'''[[19ed11/9]]''' ||'''0.44''' ||[[19ed6/5]] || 1.12

| [[20edt|20ed3/1]] || 36.27 || [[20edo|20ed2/1]] || 18.04 || [[20ed5/3]] || 11.65 || '''[[20edf|20ed3/2]]''' ||'''0.24''' ||[[20ed7/5]] || 4.74 || [[20ed4/3]] || 7.05 || [[20ed9/7]] || 8.12 || [[20ed5/4]] || 8.56 || [[20ed11/9]] || 8.67 || [[20ed6/5]] || 7.18

| [[21edt|21ed3/1]] || 22.60 || [[21edo|21ed2/1]] || 16.24 || [[21ed5/3]] || 7.30 || [[21edf|21ed3/2]] || 14.80 || [[21ed7/5]] || 10.52 || [[21ed4/3]] || 5.99 || [[21ed9/7]] || 3.27 || [[21ed5/4]] || 1.56 || '''[[21ed11/9]]''' ||'''0.43''' ||'''[[21ed6/5]]''' ||'''0.33'''

| [[23edt|23ed3/1]] || 40.41 || [[23edo|23ed2/1]] || 23.69 || [[23ed5/3]] || 1.81 || [[23edf|23ed3/2]] || 10.44 || [[23ed7/5]] || 11.72 || [[23ed4/3]] || 7.02 || [[23ed9/7]] || 4.17 || [[23ed5/4]] || 2.36 || [[23ed11/9]] || 1.15 || '''[[23ed6/5]]''' ||'''0.32'''

| [[24edt|24ed3/1]] || 11.28 || [[24edo|24ed2/1]] || 1.96 || [[24ed5/3]] || 17.83 || [[24edf|24ed3/2]] || 6.09 || '''[[24ed7/5]]''' ||'''0.11''' ||[[24ed4/3]] || 2.90 || [[24ed9/7]] || 4.50 || [[24ed5/4]] || 5.34 || [[24ed11/9]] || 5.78 || [[24ed6/5]] || 5.97

| [[28edt|28ed3/1]] || 22.69 || [[28edo|28ed2/1]] || 16.24 || [[28ed5/3]] || 7.30 || [[28edf|28ed3/2]] || 10.27 || [[28ed7/5]] || 3.58 || '''[[28ed4/3]]''' ||'''0.06''' ||[[28ed9/7]] || 1.91 || [[28ed5/4]] || 3.04 || [[28ed11/9]] || 3.71 || [[28ed6/5]] || 4.09

| [[32edt|32ed3/1]] || 11.28 || [[32edo|32ed2/1]] || 10.54 || [[32ed5/3]] || 0.59 || [[32edf|32ed3/2]] || 8.54 || [[32ed7/5]] || 6.18 || [[32ed4/3]] || 2.28 || '''[[32ed9/7]]''' ||'''0.04''' ||[[32ed5/4]] || 1.32 || [[32ed11/9]] || 2.16 || [[32ed6/5]] || 2.68

| [[36edt|36ed3/1]] || 15.14 || [[36edo|36ed2/1]] || 1.96 || [[36ed5/3]] || 6.73 || [[36edf|36ed3/2]] || 3.66 || [[36ed7/5]] || 7.98 || [[36ed4/3]] || 4.01 || [[36ed9/7]] || 1.55 || '''[[36ed5/4]]''' ||'''0.02''' ||[[36ed11/9]] || 0.95 || [[36ed6/5]] || 1.58

| [[40edt|40ed3/1]] || 11.28 || [[40edo|40ed2/1]] || 11.96 || [[40ed5/3]] || 10.46 || [[40edf|40ed3/2]] || 0.24 || [[40ed7/5]] || 4.74 || [[40ed4/3]] || 5.40 || [[40ed9/7]] || 2.75 || [[40ed5/4]] || 1.10 || '''[[40ed11/9]]''' ||'''0.01''' ||[[40ed6/5]] || 0.71

| [[44edt|44ed3/1]] || 10.34 || [[44edo|44ed2/1]] || 7.14 || [[44ed5/3]] || 4.43 || [[44edf|44ed3/2]] || 3.43 || [[44ed7/5]] || 2.09 || [[44ed4/3]] || 4.79 || [[44ed9/7]] || 3.74 || [[44ed5/4]] || 1.97 || [[44ed11/9]] || 0.80 || '''[[44ed6/5]]''' ||'''0.01'''