Thema: Minimale Potenzendifferenzen mit Potenzzahlen d^k = p^m - q^n Main index: http://www.fermatquotient.com/ Allgemeiner Hinweis: Die unendlich oft vorkommenden Null-Differenzen sind hier nicht gemeint, wie z. B. 4^3 - 8^2 = 0, sondern nur deren Abweichung. Das Resultat der von 2^120 (1.33E+36) nach unten gefundenen Minima: 1130761308917410575^2 - 1085377030725^3 = 1059502500 = 32550^2 , relativ 0.051155 253908089296698567^2 - 400975365162^3 = 868421961 = 29469^2 , relativ 0.098447 80102256650^3 - 22670814680787970^2 = 857904100 = 29290^2 , relativ 0.38677 2616752009138280^2 - 18989274720^3 = 703310400 = 26520^2 , relativ 1.08892 327094001142285^2 - 4747318680^3 = 10989225 = 3315^2 , relativ 0.055830 110633365^3 - 1163668244595^2 = 1188100 = 1090^2 , relativ 0.15139 990100714917^2 - 99338954^3 = 442225 = 665^2 , relativ 0.061797 18618529710^2 - 7024740^3 = 260100 = 510^2 , relativ 0.35205 16499304801^2 - 6481026^3 = 164025 = 405^2 , relativ 0.23788 4888682904^2 - 2880456^3 = 14400 = 120^2 , relativ 0.041849 611085363^2 - 720114^3 = 225 = 15^2 , relativ 0.0021456 137190^2 - 2660^3 = 100 = 10^2 , relativ 0.11598 65^3 - 524^2 = 49 = 7^2 , relativ 1.3687 # 312^2 - 46^3 = 8 = 2^3 , relativ 0.30052 5^3 - 11^2 = 4 = 2^2 , relativ 1.0162 # 3^2 - 2^3 = 1 , relativ 0.55204 # relativ = Differenz / Subtrahend^(1/3.5) < 2 # nicht kürzbar ggT(p,q,d) = 1 Weitere gefundene interessante Nicht-Minima: 3244590877572^3 - 2417519340^4 = 25447572^3 1712043216^4 - 2048108519592^3 = 21949272^3 609288288361312032^2 - 718701503706^3 = 24102^3 525577455216^3 - 381026421216645120^2 = 34416^3 441631056119^3 - 293487364953330240^2 = 28919^3 273820216926805962^2 - 421674177792^3 = 122934^2 , relativ 1.64088 417559118770^3 - 269821752395985190^2 = 96870^2 , relativ 1.02746 364984343900^3 - 220501401167040000^2 = 23900^3 329372813902^3 - 189030387726020071^2 = 20823^3 180529863218166528^2 - 319422890536^3 = 10712^3 295637318559^3 - 160745521450772160^2 = 19359^3 274758382273^3 - 144021200268781241^2 = 1361844^2 # 257266368125^3 - 130489186401551185^2 = 2827048900 = 53170^2 , relativ 0.46882 127201482060651600^2 - 252926760130^3 = 28670^3 116621451521322625^2 - 238699931856^3 = 116353^2 233589980096^3 - 112896717397524480^2 = 15296^3 83333290980700183^2 - 190785642450^3 = 1930108489 = 43933^2 , relativ 0.41356 178842328511^3 - 75631980600294720^2 = 11711^3 143433858568^3 - 233071280^4 = 5378568^3 131394363804^3 - 47628302652080640^2 = 8604^3 44542364312928156^2 - 125655695160^3 = 120156^2 37107458013156636^2 - 111251908469^3 = 17083^3 91586127425^3 - 27716892915033916^2 = 262087^2 # 91246085975^3 - 27562675145880000^2 = 5975^3 22566232902270816^2 - 79855722634^3 = 2678^3 58397495024^3 - 14112089674690560^2 = 3824^3 12978354281337480^2 - 55226359655^3 = 1196814025 = 34595^2 , relativ 0.74212 6700268379998841^2 - 35540876373^3 = 75858^2 32848590951^3 - 5953537831510080^2 = 2151^3 3952884900459747^2 - 25000159485^3 = 42378^2 19726764301^3 - 2770663499393374^2 = 364905^2 # 2761511922508130^2 - 19683301660^3 = 30030^2 , relativ 1.35394 2150819846277703^2 - 16662341202^3 = 450349^2 # 1791069422655997^2 - 14748288112^3 = 32259^2 # 14599373756^3 - 1764011209336320^2 = 956^3 , relativ 1.69468 1473852105862867^2 - 12950979150^3 = 293383^2 # 7210095317^3 - 612225633326387^2 = 32338^2 6575588101^3 - 533214439150635^2 = 70226^2 # 259158161262203^2 - 4064838169^3 = 29820^2 3649843439^3 - 220501401167040^2 = 13651919 = 239^3 , relativ 0.086888 # 169890020825518^2 - 3067457680^3 = 23982^2 2293904218^3 - 109865900566104^2 = 4906^3 31085203006000^2 - 988634599^3 = 16899^2 18662172009050^2 - 703571316^3 = 25020004 = 5002^2 , relativ 0.65295 16852479636189^2 - 657318528^3 = 11309769 = 3363^2 , relativ 0.31287 11115128160738^2 - 498053867^3 = 19439281 = 4409^2 , relativ 0.68214 3269539^4 - 485268420^3 = 172081^3 484580462^3 - 3266062^4 = 171898^3 472105985^3 - 10257918852944^2 = 18817^2 # 442533460^3 - 9309345956760^2 = 8720^2 7920805719336^2 - 397355816^3 = 5320^2 , relativ 1.20531 4861980948255^2 - 286995774^3 = 4899^2 , relativ 1.35085 2798198614520^2 - 198572464^3 = 7416^2 2062413100125^2 - 162025650^3 = 15^8 1066294102104^2 - 104372160^3 = 1956^3 77055139^3 - 676398151884^2 = 1267^3 445971981738^2 - 58372064^3 = 2102500 = 1450^2 , relativ 0.46343 398123925056^2 - 54118471^3 = 3795^2 349774826815^2 - 49643116^3 = 859329 = 927^2 , relativ 0.21763 238993072940^2 - 38511480^3 = 2540^2 33895685^3 - 197340680981^2 = 5042^2 # 133286762763^2 - 26093040^3 = 489^3 131994438408^2 - 25924104^3 = 3240^2 333522^4 - 23129486^3 = 26470^3 76470179302^2 - 22563^5 = 109^4 76385670375^2 - 18002850^3 = 1875^2 39109463232^2 - 11521824^3 = 921600 = 960^2 , relativ 0.81623 8982638^3 - 26921908692^2 = 602^3 7623313^3 - 21048232254^2 = 661^3 3411791^3 - 6301930488^2 = 15503^3 # 2433601^3 - 3796418340^2 = 1351^2 # 1895454^3 - 51084^4 = 13662^3 2011728813^2 - 1593601^3 = 9482^3 # 2004134496^2 - 1589588^3 = 364^3 1546629^3 - 1923442570^2 = 329^3 1212602930^2 - 1137136^3 = 462^2 , relativ 1.37590 1139917^3 - 1217053992^2 = 1549^3 # 512096003^2 - 640080^3 = 64009 = 253^2 , relativ 0.67526 # 543026^3 - 400157475^2 = 1151^3 # 250516812^2 - 397397^3 = 91^3 85016415^2 - 193346^3 = 683^2 # 77716821^2 - 182114^3 = 913^3 # 174725^3 - 73035259^2 = 362^2 # 129313^3 - 46501101^2 = 766^3 # 97338^3 - 30368520^2 = 138^3 23059835^2 - 81016^3 = 15129 = 123^2 , relativ 0.93856 21063928^2 - 76271^3 = 17^7 # 8560448^2 - 41847^3 = 8281 = 91^2 , relativ 0.90502 8097999^2 - 40326^3 = 195^2 4777461^2 - 28366^3 = 65^3 1742059^2 - 14478^3 = 127^2 13009^3 - 1483755^2 = 334^3 # 12545^3 - 1405096^2 = 97^2 # 54^7 - 10935^3 = 3^22 10685^3 - 1104490^2 = 95^2 1097520^2 - 10640^3 = 80^2 894257^2 - 9282^3 = 91^2 793084^2 - 8568^3 = 68^2 , relativ 1.96779 8138^3 - 734136^2 = 26^3 7588^3 - 813^4 = 271^3 500047^2 - 6300^3 = 47^2 , relativ 1.22354 # 205485^2 - 3482^3 = 193^3 # 78670^2 - 1836^3 = 62^2 65023^2 - 1617^3 = 14^4 1456^3 - 1455^3 = 2521^2 # 48723^2 - 1334^3 = 55^2 # 47958^2 - 1320^3 = 42^2 40380^2 - 1177^3 = 23^3 # 1153^3 - 39151^2 = 6^5 # 901^3 - 27045^2 = 26^2 , relativ 1.98301 # 443^3 - 9324^2 = 11^3 # 6831^2 - 360^3 = 3^8 6576^2 - 351^3 = 15^2 , relativ 1.48078 6159^2 - 336^3 = 15^2 , relativ 1.53727 4930^2 - 30^5 = 70^2 193^3 - 2681^2 = 6^4 # 2496^2 - 184^3 = 2^9 2415^2 - 180^3 = 15^2 154^3 - 1911^2 = 7^3 770^2 - 84^3 = 14^2 689^2 - 78^3 = 13^2 650^2 - 75^3 = 5^4 499^2 - 12^5 = 13^2 # 10^5 - 316^2 = 12^2 253^2 - 40^3 = 9 = 3^2 , relativ 0.38111 # 34^3 - 198^2 = 100 = 10^2 6^5 - 88^2 = 32 = 2^5 13^3 - 46^2 = 81 = 3^4 # 42^2 - 12^3 = 36 = 6^2 5^4 - 24^2 = 49 = 7^2 # 3^5 - 6^3 = 27 = 3^3 15^2 - 6^3 = 9 = 3^2 , relativ 1.9376 6^2 - 2^5 = 4 = 2^2 , relativ 1.4860 Nach meinen theoretischen Überlegungen, gibt es sehr wahrscheinlich keine Potenzendifferenzen kleiner als 10^6 bei Potenzen ab 2 oberhalb von 1.33E+36. Die Minima dürften etwa parallel zur 3.5 Wurzel der Zwischenergebnisse zunehmen. 06.01.2008 Richard Fischer