S-expression: Difference between revisions

3. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''² * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)²}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and S({{nowrap|''k''+1}}). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {{nowrap|{S16, S17} → {{(}}S16 * S17, S16/S17, S16² * S17, S16 * S17²{{)}} }}, and any of the two commas in the latter set imply all the other commas too.)

{| class="wikitable center-all

|-

! S-expression

! Cube Relation

|-

| S2/S3 = ([[4/3]])/([[9/8]])

| ([[4/1]])/([[3/2]])³

| [[32/27]]

|-

| S3/S4 = ([[9/8]])/([[16/15]])

| ([[5/2]])/([[4/3]])³

| [[135/128]]

|-

| S4/S5 = ([[16/15]])/([[25/24]])

| ([[2/1]])/([[5/4]])³

| [[128/125]]

|-

| S5/S6 = ([[25/24]])/([[36/35]])

| ([[7/4]])/([[6/5]])³

| [[875/864]]

|-

| S6/S7 = ([[36/35]])/([[49/48]])

| ([[8/5]])/([[7/6]])³

| [[1728/1715]]

|-

| S7/S8 = ([[49/48]])/([[64/63]])

| ([[3/2]])/([[8/7]])³

| [[1029/1024]]

|-

| S8/S9 = ([[64/63]])/([[81/80]])

| ([[10/7]])/([[9/8]])³

| [[5120/5103]]

|-

| S9/S10 = ([[81/80]])/([[100/99]])

| ([[11/8]])/([[10/9]])³

| [[8019/8000]]

|-

| S10/S11 = ([[100/99]])/([[121/120]])

| ([[4/3]])/([[11/10]])³

| [[4000/3993]]

|-

| S11/S12 = ([[121/120]])/([[144/143]])

| ([[13/10]])/([[12/11]])³

| [[17303/17280]]

|-

| S12/S13 = ([[144/143]])/([[169/168]])

| ([[14/11]])/([[13/12]])³

| [[24192/24167]]

|-

| S13/S14 = ([[169/168]])/([[196/195]])

| ([[5/4]])/([[14/13]])³

| [[10985/10976]]

|-

| S14/S15 = ([[196/195]])/([[225/224]])

| ([[16/13]])/([[15/14]])³

| [[43904/43875]]

|-

| S15/S16 = ([[225/224]])/([[256/255]])

| ([[17/14]])/([[16/15]])³

| [[57375/57344]]

|-

| S16/S17 = ([[256/255]])/([[289/288]])

| ([[6/5]])/([[17/16]])³

| [[24576/24565]]

|-

| S17/S18 = ([[289/288]])/([[324/323]])

| ([[19/16]])/([[18/17]])³

| [[93347/93312]]

|-

| S18/S19 = ([[324/323]])/([[361/360]])

| ([[20/17]])/([[19/18]])³

| [[116640/116603]]

|-

| S19/S20 = ([[361/360]])/([[400/399]])

| ([[7/6]])/([[20/19]])³

| [[48013/48000]]

|-

| S20/S21 = ([[400/399]])/([[441/440]])

| ([[22/19]])/([[21/20]])³

| [[176000/175959]]

|-

| S21/S22 = ([[441/440]])/([[484/483]])

| ([[23/20]])/([[22/21]])³

| [[213003/212960]]

|-

| S22/S23 = ([[484/483]])/([[529/528]])

| ([[8/7]])/([[23/22]])³

| [[85184/85169]]

|-

| S23/S24 = ([[529/528]])/([[576/575]])

| ([[25/22]])/([[24/23]])³

| [[304175/304128]]

|-

| S24/S25 = ([[576/575]])/([[625/624]])

| ([[26/23]])/([[25/24]])³

| [[359424/359375]]

|-

| S25/S26 = ([[625/624]])/([[676/675]])

| ([[9/8]])/([[26/25]])³

| [[140625/140608]]

|-

| S26/S27 = ([[676/675]])/([[729/728]])

| ([[28/25]])/([[27/26]])³

| [[492128/492075]]

|-

| S27/S28 = ([[729/728]])/([[784/783]])

| ([[29/26]])/([[28/27]])³

| [[570807/570752]]

|-

| S28/S29 = ([[784/783]])/([[841/840]])

| ([[10/9]])/([[29/28]])³

| [[219520/219501]]

|-

| S31/S32 = ([[961/960]])/([[1024/1023]])

| ([[11/10]])/([[32/31]])³

| [[327701/327680]]

|-

| S33/S34 = ([[1089/1088]])/([[1156/1155]])

| ([[35/32]])/([[34/33]])³

| [[1257795/1257728]]

|-

| S34/S35 = ([[1156/1155]])/([[1225/1224]])

| ([[12/11]])/([[35/34]])³

| [[471648/471625]]

|-

| S37/S38 = ([[1369/1368]])/([[1444/1443]])

| ([[13/12]])/([[38/37]])³

| [[658489/658464]]

|-

| S40/S41 = ([[1600/1599]])/([[1681/1680]])

| ([[14/13]])/([[41/40]])³

| [[896000/895973]]

|-

| S43/S44 = ([[1849/1848]])/([[1936/1935]])

| ([[15/14]])/([[44/43]])³

| [[1192605/1192576]]

|-

| S46/S47 = ([[2116/2115]])/([[2209/2208]])

| ([[16/15]])/([[47/46]])³

| [[1557376/1557345]]

|-

| S49/S50 = ([[2401/2400]])/([[2500/2499]])

| ([[17/16]])/([[50/49]])³

| [[2000033/2000000]]

|-

| S50/S51 = ([[2500/2499]])/([[2601/2600]])

| ([[52/49]])/([[51/50]])³

| [[6500000/6499899]]

|-

| S55/S56 = ([[3025/3024]])/([[3136/3135]])

| ([[19/18]])/([[56/55]])³

| [[3161125/3161088]]

|-

| S64/S65 = ([[4096/4095]])/([[4225/4224]])

| ([[22/21]])/([[65/64]])³

| [[5767168/5767125]]

|}