Magma V2.27-4 Sat Nov 11 2023 15:54:16 on euclid [Seed = 19631497] +-------------------------------------------------------------------+ | This copy of Magma has been made available through a | | generous initiative of the | | | | Simons Foundation | | | | covering U.S. Colleges, Universities, Nonprofit Research entities,| | and their students, faculty, and staff | +-------------------------------------------------------------------+ Type ? for help. Type -D to quit. Loading file "small-degree-exponents.mag" Outputting values claimed in Theorem 5.9. delta_{d,20} = 11/2128/(d + 41/3192) delta_{d,85} = 24911/99008/(d + 943/74256) delta_{1,20} = 33/6466 = 0.00510361892978657593566347046088 delta_{1,23} = 475/18908 = 0.0251216416331711444891051406812 delta_{1,30} = 76/1213 = 0.0626545754328112118713932399011 delta_{1,50} = 143/991 = 0.144298688193743693239152371342 delta_{1,70} = 4707/22721 = 0.207165177589014568020773733551 delta_{1,85} = 74733/300796 = 0.248450777270974348063139137489 Verifying the claim that the sequence delta_{d,n} for fixed d is increasing: Each successive difference delta_{d,n+1}-delta_{d,n} has no positive root in d. Each successive difference is positive when d=1. Every successive difference is positive, so the claim holds. Combined with the provided value of delta_{d,20}, this shows that every delta_{d,n} is positive. Verifying the claim that every gamma_{d,n} is positive: Every gamma is the reciprocal of a positive linear polynomial. Every gamma is positive when d=1. This verifies the claim that each gamma is positive. ----------------- For reference, we now print all values of delta when d=1. delta_{1, 20 } = 33/6466 = 0.00510361892978657593566347046088 delta_{1, 21 } = 23/1844 = 0.0124728850325379609544468546638 delta_{1, 22 } = 3/149 = 0.0201342281879194630872483221476 delta_{1, 23 } = 475/18908 = 0.0251216416331711444891051406812 delta_{1, 24 } = 65/2126 = 0.0305738476011288805268109125117 delta_{1, 25 } = 27/742 = 0.0363881401617250673854447439353 delta_{1, 26 } = 28/659 = 0.0424886191198786039453717754173 delta_{1, 27 } = 1421/29108 = 0.0488181943108423801016902569740 delta_{1, 28 } = 285/5366 = 0.0531121878494222884830413715990 delta_{1, 29 } = 1023/17716 = 0.0577444118311131180853465793633 delta_{1, 30 } = 76/1213 = 0.0626545754328112118713932399011 delta_{1, 31 } = 957/14116 = 0.0677954094644375177103995466137 delta_{1, 32 } = 1683/23014 = 0.0731293994959589814895281133224 delta_{1, 33 } = 37/476 = 0.0777310924369747899159663865546 delta_{1, 34 } = 41/502 = 0.0816733067729083665338645418327 delta_{1, 35 } = 37/431 = 0.0858468677494199535962877030162 delta_{1, 36 } = 95/1053 = 0.0902184235517568850902184235518 delta_{1, 37 } = 2145/22636 = 0.0947605584025446191906697296343 delta_{1, 38 } = 181/1820 = 0.0994505494505494505494505494506 delta_{1, 39 } = 8077/78004 = 0.103545972001435823803907491923 delta_{1, 40 } = 1491/13922 = 0.107096681511277115356988938371 delta_{1, 41 } = 43/388 = 0.110824742268041237113402061856 delta_{1, 42 } = 124/1081 = 0.114708603145235892691951896392 delta_{1, 43 } = 445/3748 = 0.118729989327641408751334044824 delta_{1, 44 } = 6601/53722 = 0.122873310747924500204757827333 delta_{1, 45 } = 329/2588 = 0.127125193199381761978361669243 delta_{1, 46 } = 123/944 = 0.130296610169491525423728813559 delta_{1, 47 } = 349/2612 = 0.133614088820826952526799387443 delta_{1, 48 } = 1855/13534 = 0.137062213684054972661445249002 delta_{1, 49 } = 10047/71444 = 0.140627624433122445551760819663 delta_{1, 50 } = 143/991 = 0.144298688193743693239152371342 delta_{1, 51 } = 23479/158572 = 0.148065232197361450949726307293 delta_{1, 52 } = 12663/83354 = 0.151918324255584615015476161912 delta_{1, 53 } = 1045/6748 = 0.154860699466508595139300533491 delta_{1, 54 } = 364/2305 = 0.157917570498915401301518438178 delta_{1, 55 } = 31179/193564 = 0.161078506333822404992663925110 delta_{1, 56 } = 3335/20294 = 0.164334285995860845570119247068 delta_{1, 57 } = 8909/53132 = 0.167676729654445531882857788150 delta_{1, 58 } = 4755/27791 = 0.171098557086826670504839696305 delta_{1, 59 } = 8113/46468 = 0.174593268485839717655160540587 delta_{1, 60 } = 3596/20233 = 0.177729451885533534325112440073 delta_{1, 61 } = 6543/36244 = 0.180526431961152190707427436265 delta_{1, 62 } = 3036/16553 = 0.183410862079381381018546487042 delta_{1, 63 } = 2145/11509 = 0.186375879746285515683378225736 delta_{1, 64 } = 1947/10279 = 0.189415312773616110516587216655 delta_{1, 65 } = 57687/299636 = 0.192523595295625358768639282329 delta_{1, 66 } = 391/1998 = 0.195695695695695695695695695696 delta_{1, 67 } = 32223/161984 = 0.198927054523903595416831291979 delta_{1, 68 } = 6797/33674 = 0.201847122408980222129833105660 delta_{1, 69 } = 1491/7292 = 0.204470652770159078442128359846 delta_{1, 70 } = 4707/22721 = 0.207165177589014568020773733551 delta_{1, 71 } = 2263/10780 = 0.209925788497217068645640074212 delta_{1, 72 } = 13875/65218 = 0.212748014351866049250206998068 delta_{1, 73 } = 8745/40556 = 0.215627773942203373113719301706 delta_{1, 74 } = 11476/52507 = 0.218561334679185632391871559982 delta_{1, 75 } = 32109/144932 = 0.221545276405486711009300913532 delta_{1, 76 } = 10101/44978 = 0.224576459602472319800791498066 delta_{1, 77 } = 52851/232804 = 0.227019295201113382931564749747 delta_{1, 78 } = 3455/15053 = 0.229522354347970504218428220288 delta_{1, 79 } = 115587/498044 = 0.232081904410052123908730955498 delta_{1, 80 } = 60393/257326 = 0.234694512019772584192813784849 delta_{1, 81 } = 1577/6644 = 0.237357013847080072245635159542 delta_{1, 82 } = 16464/68581 = 0.240066490719003805718785086248 delta_{1, 83 } = 137445/566036 = 0.242820244648750256167452246854 delta_{1, 84 } = 71681/291842 = 0.245615778400641442972567348086 delta_{1, 85 } = 74733/300796 = 0.248450777270974348063139137489 ---------------- We also now print all values of delta and gamma. delta_{d, 20 } = 11/2128/(d + 41/3192) gamma_{d, 20 } = 1969/3192/(d + 41/3192) delta_{d, 21 } = 23/1820/(d + 6/455) gamma_{d, 21 } = 161/260/(d + 6/455) delta_{d, 22 } = 1/49/(d + 2/147) gamma_{d, 22 } = 61/98/(d + 2/147) delta_{d, 23 } = 475/18656/(d + 63/4664) gamma_{d, 23 } = 11625/18656/(d + 63/4664) delta_{d, 24 } = 325/10488/(d + 71/5244) gamma_{d, 24 } = 273/437/(d + 71/5244) delta_{d, 25 } = 9/244/(d + 5/366) gamma_{d, 25 } = 153/244/(d + 5/366) delta_{d, 26 } = 14/325/(d + 9/650) gamma_{d, 26 } = 63/100/(d + 9/650) delta_{d, 27 } = 1421/28704/(d + 101/7176) gamma_{d, 27 } = 6061/9568/(d + 101/7176) delta_{d, 28 } = 95/1764/(d + 37/2646) gamma_{d, 28 } = 40/63/(d + 37/2646) delta_{d, 29 } = 341/5824/(d + 61/4368) gamma_{d, 29 } = 11129/17472/(d + 61/4368) delta_{d, 30 } = 304/4785/(d + 67/4785) gamma_{d, 30 } = 204/319/(d + 67/4785) delta_{d, 31 } = 11/160/(d + 49/3480) gamma_{d, 31 } = 2981/4640/(d + 49/3480) delta_{d, 32 } = 561/7564/(d + 161/11346) gamma_{d, 32 } = 7327/11346/(d + 161/11346) delta_{d, 33 } = 111/1408/(d + 5/352) gamma_{d, 33 } = 83/128/(d + 5/352) delta_{d, 34 } = 41/495/(d + 7/495) gamma_{d, 34 } = 322/495/(d + 7/495) delta_{d, 35 } = 37/425/(d + 6/425) gamma_{d, 35 } = 111/170/(d + 6/425) delta_{d, 36 } = 57/623/(d + 44/3115) gamma_{d, 36 } = 817/1246/(d + 44/3115) delta_{d, 37 } = 143/1488/(d + 79/5580) gamma_{d, 37 } = 4901/7440/(d + 79/5580) delta_{d, 38 } = 362/3589/(d + 51/3589) gamma_{d, 38 } = 2376/3589/(d + 51/3589) delta_{d, 39 } = 8077/76912/(d + 273/19228) gamma_{d, 39 } = 51127/76912/(d + 273/19228) delta_{d, 40 } = 497/4576/(d + 97/6864) gamma_{d, 40 } = 763/1144/(d + 97/6864) delta_{d, 41 } = 989/8800/(d + 31/2200) gamma_{d, 41 } = 5891/8800/(d + 31/2200) delta_{d, 42 } = 62/533/(d + 15/1066) gamma_{d, 42 } = 1433/2132/(d + 15/1066) delta_{d, 43 } = 445/3696/(d + 13/924) gamma_{d, 43 } = 2495/3696/(d + 13/924) delta_{d, 44 } = 943/7568/(d + 373/26488) gamma_{d, 44 } = 1633/2408/(d + 373/26488) delta_{d, 45 } = 329/2552/(d + 9/638) gamma_{d, 45 } = 1739/2552/(d + 9/638) delta_{d, 46 } = 656/4965/(d + 209/14895) gamma_{d, 46 } = 3394/4965/(d + 209/14895) delta_{d, 47 } = 349/2576/(d + 9/644) gamma_{d, 47 } = 1767/2576/(d + 9/644) delta_{d, 48 } = 1855/13348/(d + 93/6674) gamma_{d, 48 } = 4595/6674/(d + 93/6674) delta_{d, 49 } = 3349/23488/(d + 245/17616) gamma_{d, 49 } = 16235/23488/(d + 245/17616) delta_{d, 50 } = 2717/18571/(d + 258/18571) gamma_{d, 50 } = 25779/37142/(d + 258/18571) delta_{d, 51 } = 23479/156400/(d + 543/39100) gamma_{d, 51 } = 6413/9200/(d + 543/39100) delta_{d, 52 } = 4221/27404/(d + 571/41106) gamma_{d, 52 } = 369/527/(d + 571/41106) delta_{d, 53 } = 1045/6656/(d + 23/1664) gamma_{d, 53 } = 4675/6656/(d + 23/1664) delta_{d, 54 } = 280/1749/(d + 313/22737) gamma_{d, 54 } = 5341/7579/(d + 313/22737) delta_{d, 55 } = 10393/63648/(d + 655/47736) gamma_{d, 55 } = 45011/63648/(d + 655/47736) delta_{d, 56 } = 667/4004/(d + 137/10010) gamma_{d, 56 } = 203/286/(d + 137/10010) delta_{d, 57 } = 8909/52416/(d + 179/13104) gamma_{d, 57 } = 12449/17472/(d + 179/13104) delta_{d, 58 } = 1585/9139/(d + 374/27417) gamma_{d, 58 } = 13075/18278/(d + 374/27417) delta_{d, 59 } = 2135/12064/(d + 781/57304) gamma_{d, 59 } = 164639/229216/(d + 781/57304) delta_{d, 60 } = 372/2065/(d + 814/59885) gamma_{d, 60 } = 17267/23954/(d + 814/59885) delta_{d, 61 } = 2181/11920/(d + 121/8940) gamma_{d, 61 } = 8619/11920/(d + 121/8940) delta_{d, 62 } = 4048/21777/(d + 881/65331) gamma_{d, 62 } = 47392/65331/(d + 881/65331) delta_{d, 63 } = 6435/34069/(d + 458/34069) gamma_{d, 63 } = 7085/9734/(d + 458/34069) delta_{d, 64 } = 649/3381/(d + 136/10143) gamma_{d, 64 } = 4939/6762/(d + 136/10143) delta_{d, 65 } = 2747/14080/(d + 989/73920) gamma_{d, 65 } = 43349/59136/(d + 989/73920) delta_{d, 66 } = 1173/5915/(d + 79/5915) gamma_{d, 66 } = 4352/5915/(d + 79/5915) delta_{d, 67 } = 10741/53284/(d + 533/39963) gamma_{d, 67 } = 39353/53284/(d + 533/39963) delta_{d, 68 } = 6797/33232/(d + 221/16616) gamma_{d, 68 } = 12313/16616/(d + 221/16616) delta_{d, 69 } = 4473/21590/(d + 143/10795) gamma_{d, 69 } = 8023/10795/(d + 143/10795) delta_{d, 70 } = 1569/7475/(d + 296/22425) gamma_{d, 70 } = 2229/2990/(d + 296/22425) delta_{d, 71 } = 2263/10640/(d + 1/76) gamma_{d, 71 } = 7957/10640/(d + 1/76) delta_{d, 72 } = 8325/38624/(d + 1267/96560) gamma_{d, 72 } = 36223/48280/(d + 1267/96560) delta_{d, 73 } = 2915/13344/(d + 131/10008) gamma_{d, 73 } = 10045/13344/(d + 131/10008) delta_{d, 74 } = 5738/25915/(d + 677/51830) gamma_{d, 74 } = 78299/103660/(d + 677/51830) delta_{d, 75 } = 96327/429200/(d + 1399/107300) gamma_{d, 75 } = 13013/17168/(d + 1399/107300) delta_{d, 76 } = 91/400/(d + 289/22200) gamma_{d, 76 } = 5629/7400/(d + 289/22200) delta_{d, 77 } = 17617/76608/(d + 745/57456) gamma_{d, 77 } = 25043/32832/(d + 745/57456) delta_{d, 78 } = 3455/14861/(d + 192/14861) gamma_{d, 78 } = 22735/29722/(d + 192/14861) delta_{d, 79 } = 38529/163904/(d + 1583/122928) gamma_{d, 79 } = 125739/163904/(d + 1583/122928) delta_{d, 80 } = 20131/84688/(d + 1631/127032) gamma_{d, 80 } = 12218/15879/(d + 1631/127032) delta_{d, 81 } = 1577/6560/(d + 21/1640) gamma_{d, 81 } = 5063/6560/(d + 21/1640) delta_{d, 82 } = 1372/5643/(d + 865/67716) gamma_{d, 82 } = 34951/45144/(d + 865/67716) delta_{d, 83 } = 45815/186304/(d + 1781/139728) gamma_{d, 83 } = 434095/558912/(d + 1781/139728) delta_{d, 84 } = 71681/288176/(d + 1833/144088) gamma_{d, 84 } = 16039/20584/(d + 1833/144088) delta_{d, 85 } = 24911/99008/(d + 943/74256) gamma_{d, 85 } = 4553/5824/(d + 943/74256) Total time: 0.030 seconds, Total memory usage: 32.09MB