Для перевода 7-ого числа 123243431245633564343654521313303116130235241205652132626625412213261201202556455053034566542165432600005546660561650624603035242216053203436156133301552653011 в десятичное необходимо записать его в виде многочлена, состоящего из произведений цифр вида:
A7 = an-1 ∙ 7n-1 + an-2 ∙ 7n-2 + ∙∙∙ + a0 ∙ 70
В результате преобразований получим:
1232434312456335643436545213133031161302352412056521326266254122132612012025564550530345665421654326000055466605616506246030352422160532034361561333015526530117=1 ∙ 7158 + 2 ∙ 7157 + 3 ∙ 7156 + 2 ∙ 7155 + 4 ∙ 7154 + 3 ∙ 7153 + 4 ∙ 7152 + 3 ∙ 7151 + 1 ∙ 7150 + 2 ∙ 7149 + 4 ∙ 7148 + 5 ∙ 7147 + 6 ∙ 7146 + 3 ∙ 7145 + 3 ∙ 7144 + 5 ∙ 7143 + 6 ∙ 7142 + 4 ∙ 7141 + 3 ∙ 7140 + 4 ∙ 7139 + 3 ∙ 7138 + 6 ∙ 7137 + 5 ∙ 7136 + 4 ∙ 7135 + 5 ∙ 7134 + 2 ∙ 7133 + 1 ∙ 7132 + 3 ∙ 7131 + 1 ∙ 7130 + 3 ∙ 7129 + 3 ∙ 7128 + 0 ∙ 7127 + 3 ∙ 7126 + 1 ∙ 7125 + 1 ∙ 7124 + 6 ∙ 7123 + 1 ∙ 7122 + 3 ∙ 7121 + 0 ∙ 7120 + 2 ∙ 7119 + 3 ∙ 7118 + 5 ∙ 7117 + 2 ∙ 7116 + 4 ∙ 7115 + 1 ∙ 7114 + 2 ∙ 7113 + 0 ∙ 7112 + 5 ∙ 7111 + 6 ∙ 7110 + 5 ∙ 7109 + 2 ∙ 7108 + 1 ∙ 7107 + 3 ∙ 7106 + 2 ∙ 7105 + 6 ∙ 7104 + 2 ∙ 7103 + 6 ∙ 7102 + 6 ∙ 7101 + 2 ∙ 7100 + 5 ∙ 799 + 4 ∙ 798 + 1 ∙ 797 + 2 ∙ 796 + 2 ∙ 795 + 1 ∙ 794 + 3 ∙ 793 + 2 ∙ 792 + 6 ∙ 791 + 1 ∙ 790 + 2 ∙ 789 + 0 ∙ 788 + 1 ∙ 787 + 2 ∙ 786 + 0 ∙ 785 + 2 ∙ 784 + 5 ∙ 783 + 5 ∙ 782 + 6 ∙ 781 + 4 ∙ 780 + 5 ∙ 779 + 5 ∙ 778 + 0 ∙ 777 + 5 ∙ 776 + 3 ∙ 775 + 0 ∙ 774 + 3 ∙ 773 + 4 ∙ 772 + 5 ∙ 771 + 6 ∙ 770 + 6 ∙ 769 + 5 ∙ 768 + 4 ∙ 767 + 2 ∙ 766 + 1 ∙ 765 + 6 ∙ 764 + 5 ∙ 763 + 4 ∙ 762 + 3 ∙ 761 + 2 ∙ 760 + 6 ∙ 759 + 0 ∙ 758 + 0 ∙ 757 + 0 ∙ 756 + 0 ∙ 755 + 5 ∙ 754 + 5 ∙ 753 + 4 ∙ 752 + 6 ∙ 751 + 6 ∙ 750 + 6 ∙ 749 + 0 ∙ 748 + 5 ∙ 747 + 6 ∙ 746 + 1 ∙ 745 + 6 ∙ 744 + 5 ∙ 743 + 0 ∙ 742 + 6 ∙ 741 + 2 ∙ 740 + 4 ∙ 739 + 6 ∙ 738 + 0 ∙ 737 + 3 ∙ 736 + 0 ∙ 735 + 3 ∙ 734 + 5 ∙ 733 + 2 ∙ 732 + 4 ∙ 731 + 2 ∙ 730 + 2 ∙ 729 + 1 ∙ 728 + 6 ∙ 727 + 0 ∙ 726 + 5 ∙ 725 + 3 ∙ 724 + 2 ∙ 723 + 0 ∙ 722 + 3 ∙ 721 + 4 ∙ 720 + 3 ∙ 719 + 6 ∙ 718 + 1 ∙ 717 + 5 ∙ 716 + 6 ∙ 715 + 1 ∙ 714 + 3 ∙ 713 + 3 ∙ 712 + 3 ∙ 711 + 0 ∙ 710 + 1 ∙ 79 + 5 ∙ 78 + 5 ∙ 77 + 2 ∙ 76 + 6 ∙ 75 + 5 ∙ 74 + 3 ∙ 73 + 0 ∙ 72 + 1 ∙ 71 + 1 ∙ 70 = 1 ∙ 3.3534383162004E+133 + 2 ∙ 4.7906261660005E+132 + 3 ∙ 6.843751665715E+131 + 2 ∙ 9.7767880938786E+130 + 4 ∙ 1.3966840134112E+130 + 3 ∙ 1.9952628763017E+129 + 4 ∙ 2.8503755375739E+128 + 3 ∙ 4.071965053677E+127 + 1 ∙ 5.8170929338243E+126 + 2 ∙ 8.3101327626062E+125 + 4 ∙ 1.1871618232295E+125 + 5 ∙ 1.6959454617564E+124 + 6 ∙ 2.4227792310805E+123 + 3 ∙ 3.4611131872579E+122 + 3 ∙ 4.9444474103684E+121 + 5 ∙ 7.0634963005263E+120 + 6 ∙ 1.0090709000752E+120 + 4 ∙ 1.4415298572503E+119 + 3 ∙ 2.0593283675004E+118 + 4 ∙ 2.9418976678577E+117 + 3 ∙ 4.2027109540824E+116 + 6 ∙ 6.0038727915463E+115 + 5 ∙ 8.5769611307804E+114 + 4 ∙ 1.2252801615401E+114 + 5 ∙ 1.7504002307715E+113 + 2 ∙ 2.500571758245E+112 + 1 ∙ 3.5722453689215E+111 + 3 ∙ 5.1032076698878E+110 + 1 ∙ 7.2902966712683E+109 + 3 ∙ 1.0414709530383E+109 + 3 ∙ 1.4878156471976E+108 + 0 ∙ 2.125450924568E+107 + 3 ∙ 3.0363584636686E+106 + 1 ∙ 4.337654948098E+105 + 1 ∙ 6.1966499258543E+104 + 6 ∙ 8.8523570369347E+103 + 1 ∙ 1.2646224338478E+103 + 3 ∙ 1.8066034769254E+102 + 0 ∙ 2.5808621098935E+101 + 2 ∙ 3.6869458712764E+100 + 3 ∙ 5.2670655303949E+99 + 5 ∙ 7.5243793291355E+98 + 2 ∙ 1.0749113327336E+98 + 4 ∙ 1.5355876181909E+97 + 1 ∙ 2.1936965974156E+96 + 2 ∙ 3.1338522820223E+95 + 0 ∙ 4.4769318314604E+94 + 5 ∙ 6.3956169020863E+93 + 6 ∙ 9.136595574409E+92 + 5 ∙ 1.3052279392013E+92 + 2 ∙ 1.8646113417161E+91 + 1 ∙ 2.6637304881659E+90 + 3 ∙ 3.8053292688084E+89 + 2 ∙ 5.4361846697263E+88 + 6 ∙ 7.765978099609E+87 + 2 ∙ 1.1094254428013E+87 + 6 ∙ 1.5848934897161E+86 + 6 ∙ 2.2641335567373E+85 + 2 ∙ 3.2344765096248E+84 + 5 ∙ 4.6206807280354E+83 + 4 ∙ 6.600972468622E+82 + 1 ∙ 9.4299606694599E+81 + 2 ∙ 1.3471372384943E+81 + 2 ∙ 1.9244817692775E+80 + 1 ∙ 2.7492596703965E+79 + 3 ∙ 3.9275138148521E+78 + 2 ∙ 5.6107340212173E+77 + 6 ∙ 8.0153343160247E+76 + 1 ∙ 1.1450477594321E+76 + 2 ∙ 1.6357825134744E+75 + 0 ∙ 2.3368321621063E+74 + 1 ∙ 3.3383316601519E+73 + 2 ∙ 4.7690452287884E+72 + 0 ∙ 6.8129217554121E+71 + 2 ∙ 9.7327453648744E+70 + 5 ∙ 1.3903921949821E+70 + 5 ∙ 1.9862745642601E+69 + 6 ∙ 2.8375350918001E+68 + 4 ∙ 4.0536215597144E+67 + 5 ∙ 5.7908879424492E+66 + 5 ∙ 8.2726970606417E+65 + 0 ∙ 1.181813865806E+65 + 5 ∙ 1.6883055225799E+64 + 3 ∙ 2.4118650322571E+63 + 0 ∙ 3.4455214746529E+62 + 3 ∙ 4.9221735352185E+61 + 4 ∙ 7.0316764788836E+60 + 5 ∙ 1.0045252112691E+60 + 6 ∙ 1.4350360160987E+59 + 6 ∙ 2.0500514515695E+58 + 5 ∙ 2.9286449308136E+57 + 4 ∙ 4.1837784725909E+56 + 2 ∙ 5.9768263894156E+55 + 1 ∙ 8.5383234134509E+54 + 6 ∙ 1.2197604876358E+54 + 5 ∙ 1.7425149823369E+53 + 4 ∙ 2.4893071176242E+52 + 3 ∙ 3.5561530251774E+51 + 2 ∙ 5.0802186073962E+50 + 6 ∙ 7.2574551534232E+49 + 0 ∙ 1.0367793076319E+49 + 0 ∙ 1.481113296617E+48 + 0 ∙ 2.1158761380243E+47 + 0 ∙ 3.0226801971775E+46 + 5 ∙ 4.3181145673964E+45 + 5 ∙ 6.1687350962806E+44 + 4 ∙ 8.8124787089723E+43 + 6 ∙ 1.2589255298532E+43 + 6 ∙ 1.7984650426474E+42 + 6 ∙ 2.5692357752106E+41 + 0 ∙ 3.6703368217294E+40 + 5 ∙ 5.2433383167563E+39 + 6 ∙ 7.4904833096519E+38 + 1 ∙ 1.070069044236E+38 + 6 ∙ 1.5286700631943E+37 + 5 ∙ 2.1838143759918E+36 + 0 ∙ 3.1197348228454E+35 + 6 ∙ 4.4567640326363E+34 + 2 ∙ 6.366805760909E+33 + 4 ∙ 9.0954368012986E+32 + 6 ∙ 1.2993481144712E+32 + 0 ∙ 1.8562115921018E+31 + 3 ∙ 2.6517308458597E+30 + 0 ∙ 3.7881869226566E+29 + 3 ∙ 5.4116956037952E+28 + 5 ∙ 7.7309937197074E+27 + 2 ∙ 1.1044276742439E+27 + 4 ∙ 1.5777538203485E+26 + 2 ∙ 2.2539340290692E+25 + 2 ∙ 3.2199057558132E+24 + 1 ∙ 4.5998653654474E+23 + 6 ∙ 6.5712362363534E+22 + 0 ∙ 9.3874803376478E+21 + 5 ∙ 1.341068619664E+21 + 3 ∙ 1.9158123138057E+20 + 2 ∙ 2.7368747340081E+19 + 0 ∙ 3909821048582988049 + 3 ∙ 558545864083284007 + 4 ∙ 79792266297612001 + 3 ∙ 11398895185373143 + 6 ∙ 1628413597910449 + 1 ∙ 232630513987207 + 5 ∙ 33232930569601 + 6 ∙ 4747561509943 + 1 ∙ 678223072849 + 3 ∙ 96889010407 + 3 ∙ 13841287201 + 3 ∙ 1977326743 + 0 ∙ 282475249 + 1 ∙ 40353607 + 5 ∙ 5764801 + 5 ∙ 823543 + 2 ∙ 117649 + 6 ∙ 16807 + 5 ∙ 2401 + 3 ∙ 343 + 0 ∙ 49 + 1 ∙ 7 + 1 ∙ 1 = 3.3534383162004E+133 + 9.581252332001E+132 + 2.0531254997145E+132 + 1.9553576187757E+131 + 5.5867360536449E+130 + 5.9857886289053E+129 + 1.1401502150296E+129 + 1.2215895161031E+128 + 5.8170929338243E+126 + 1.6620265525212E+126 + 4.7486472929178E+125 + 8.4797273087818E+124 + 1.4536675386483E+124 + 1.0383339561774E+123 + 1.4833342231105E+122 + 3.5317481502632E+121 + 6.0544254004511E+120 + 5.7661194290011E+119 + 6.1779851025012E+118 + 1.1767590671431E+118 + 1.2608132862247E+117 + 3.6023236749278E+116 + 4.2884805653902E+115 + 4.9011206461603E+114 + 8.7520011538576E+113 + 5.0011435164901E+112 + 3.5722453689215E+111 + 1.5309623009663E+111 + 7.2902966712683E+109 + 3.124412859115E+109 + 4.4634469415928E+108 + 0 + 9.1090753910058E+106 + 4.337654948098E+105 + 6.1966499258543E+104 + 5.3114142221608E+104 + 1.2646224338478E+103 + 5.4198104307763E+102 + 0 + 7.3738917425528E+100 + 1.5801196591185E+100 + 3.7621896645678E+99 + 2.1498226654673E+98 + 6.1423504727637E+97 + 2.1936965974156E+96 + 6.2677045640446E+95 + 0 + 3.1978084510432E+94 + 5.4819573446454E+93 + 6.5261396960065E+92 + 3.7292226834323E+91 + 2.6637304881659E+90 + 1.1415987806425E+90 + 1.0872369339453E+89 + 4.6595868597654E+88 + 2.2188508856026E+87 + 9.5093609382968E+86 + 1.3584801340424E+86 + 6.4689530192495E+84 + 2.3103403640177E+84 + 2.6403889874488E+83 + 9.4299606694599E+81 + 2.6942744769886E+81 + 3.8489635385551E+80 + 2.7492596703965E+79 + 1.1782541444556E+79 + 1.1221468042435E+78 + 4.8092005896148E+77 + 1.1450477594321E+76 + 3.2715650269489E+75 + 0 + 3.3383316601519E+73 + 9.5380904575769E+72 + 0 + 1.9465490729749E+71 + 6.9519609749103E+70 + 9.9313728213004E+69 + 1.7025210550801E+69 + 1.6214486238858E+68 + 2.8954439712246E+67 + 4.1363485303209E+66 + 0 + 8.4415276128997E+64 + 7.2355950967712E+63 + 0 + 1.4766520605655E+62 + 2.8126705915534E+61 + 5.0226260563454E+60 + 8.6102160965921E+59 + 1.2300308709417E+59 + 1.4643224654068E+58 + 1.6735113890364E+57 + 1.1953652778831E+56 + 8.5383234134509E+54 + 7.318562925815E+54 + 8.7125749116845E+53 + 9.9572284704966E+52 + 1.0668459075532E+52 + 1.0160437214792E+51 + 4.3544730920539E+50 + 0 + 0 + 0 + 0 + 2.1590572836982E+46 + 3.0843675481403E+45 + 3.5249914835889E+44 + 7.5535531791191E+43 + 1.0790790255884E+43 + 1.5415414651264E+42 + 0 + 2.6216691583782E+40 + 4.4942899857911E+39 + 1.070069044236E+38 + 9.1720203791655E+37 + 1.0919071879959E+37 + 0 + 2.6740584195818E+35 + 1.2733611521818E+34 + 3.6381747205194E+33 + 7.7960886868274E+32 + 0 + 7.955192537579E+30 + 0 + 1.6235086811386E+29 + 3.8654968598537E+28 + 2.2088553484878E+27 + 6.3110152813938E+26 + 4.5078680581385E+25 + 6.4398115116264E+24 + 4.5998653654474E+23 + 3.9427417418121E+23 + 0 + 6.7053430983198E+21 + 5.747436941417E+20 + 5.4737494680162E+19 + 0 + 1675637592249852021 + 319169065190448004 + 34196685556119429 + 9770481587462694 + 232630513987207 + 166164652848005 + 28485369059658 + 678223072849 + 290667031221 + 41523861603 + 5931980229 + 0 + 40353607 + 28824005 + 4117715 + 235298 + 100842 + 12005 + 1029 + 0 + 7 + 1 = 4.5427420268475E+13310
Таким образом:
1232434312456335643436545213133031161302352412056521326266254122132612012025564550530345665421654326000055466605616506246030352422160532034361561333015526530117 = 4.5427420268475E+13310.