Sharedwww / tables / abc / abc-table.texOpen in CoCalc
Author: William A. Stein
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\documentclass{article}
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\centerline{$c\leq 10000$}
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\begin{document}
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$$\begin{array}{llllll}
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a & b & c & c & \!\!\!\mbox{\rm cond}(\mbox{\rm rad}) &
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\displaystyle \frac{\log(c)}{\log(\mbox{\rm cond)}}\\\hline
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1& 1& 2& 2& 2& 1\\
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1& 2^{3}& 3^{2}& 9& 6& 1.2263\\
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5& 3^{3}& 2^{5}& 32& 30& 1.019\\
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1& 2^{4}\cdot3& 7^{2}& 49& 42& 1.0412\\
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1& 3^{2}\cdot7& 2^{6}& 64& 42& 1.1127\\
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1& 2^{4}\cdot5& 3^{4}& 81& 30& 1.292\\
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2^{5}& 7^{2}& 3^{4}& 81& 42& 1.1757\\
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2^{2}& 11^{2}& 5^{3}& 125& 110& 1.0272\\
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3& 5^{3}& 2^{7}& 128& 30& 1.4266\\
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1& 2^{5}\cdot7&3^{2}\cdot5^{2}& 225& 210&1.0129\\
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1& 2\cdot11^{2}& 3^{5}& 243& 66& 1.3111\\
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2& 3^{5}&5\cdot7^{2}& 245& 210& 1.0288\\
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7& 3^{5}&2\cdot5^{3}& 250& 210& 1.0326\\
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13& 3^{5}& 2^{8}& 256& 78& 1.2728\\
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3^{4}& 5^{2}\cdot7& 2^{8}& 256& 210& 1.037\\
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1& 2^{5}\cdot3^{2}& 17^{2}& 289& 102& 1.2252\\
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2^{2}\cdot5^{2}& 3^{5}& 7^{3}& 343& 210& 1.0918\\
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2^{5}& 7^{3}&3\cdot5^{3}& 375& 210& 1.1084\\
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5& 3\cdot13^{2}& 2^{9}& 512& 390& 1.0456\\
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13^{2}& 7^{3}& 2^{9}& 512& 182& 1.1988\\
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1& 2^{9}&3^{3}\cdot19& 513& 114& 1.3176\\
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3^{3}& 2^{9}&7^{2}\cdot11& 539& 462& 1.0251\\
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1& 2^{4}\cdot3\cdot13& 5^{4}& 625& 390& 1.079\\
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7^{2}& 2^{6}\cdot3^{2}& 5^{4}& 625& 210& 1.204\\
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3^{4}& 2^{5}\cdot17& 5^{4}& 625& 510& 1.0326\\
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1& 3^{3}\cdot5^{2}&2^{2}\cdot13^{2}& 676& 390& 1.0922\\
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1& 2^{3}\cdot7\cdot13& 3^{6}& 729& 546& 1.0459\\
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5^{2}& 2^{6}\cdot11& 3^{6}& 729& 330& 1.1367\\
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2^{3}\cdot13& 5^{4}& 3^{6}& 729& 390& 1.1048\\
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2^{3}\cdot5^{2}& 23^{2}& 3^{6}& 729& 690& 1.0084\\
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1& 2^{6}\cdot3\cdot5& 31^{2}& 961& 930& 1.0048\\
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7^{3}& 5^{4}&2^{3}\cdot11^{2}& 968& 770& 1.0344\\
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1& 2^{10}&5^{2}\cdot41& 1025& 410& 1.1523\\
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5& 2^{10}&3\cdot7^{3}& 1029& 210& 1.2972\\
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1& 3^{5}\cdot5&2^{6}\cdot19& 1216& 570& 1.1194\\
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2^{3}& 3^{3}\cdot7^{2}& 11^{3}& 1331& 462& 1.1725\\
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3^{4}& 2\cdot5^{4}& 11^{3}& 1331& 330& 1.2405\\
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3^{5}& 2^{6}\cdot17& 11^{3}& 1331& 1122& 1.0243\\
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2^{7}\cdot5& 3^{6}& 37^{2}& 1369& 1110& 1.0299\\
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2^{8}& 11^{3}&3\cdot23^{2}& 1587& 1518& 1.0061\\
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3^{4}& 2^{6}\cdot5^{2}& 41^{2}& 1681& 1230& 1.0439\\
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\end{array}
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$$
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\newpage
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$$\begin{array}{llllll}
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a & b & c & c & \!\!\!\mbox{\rm cond}(\mbox{\rm rad}) &
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\displaystyle \frac{\log(c)}{\log(\mbox{\rm cond)}}\\\hline
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23& 3^{4}\cdot5^{2}& 2^{11}& 2048& 690& 1.1664\\
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5^{2}& 7\cdot17^{2}& 2^{11}& 2048& 1190& 1.0767\\
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3^{5}& 5\cdot19^{2}& 2^{11}& 2048& 570& 1.2016\\
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3^{2}& 2^{11}&11^{2}\cdot17& 2057& 1122&
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1.0863\\
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11& 2^{7}\cdot17& 3^{7}& 2187& 1122& 1.095\\
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139& 2^{11}& 3^{7}& 2187& 834& 1.1433\\
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2^{9}& 5^{2}\cdot67& 3^{7}& 2187& 2010& 1.0111\\
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2\cdot5& 3^{7}& 13^{3}& 2197& 390& 1.2898\\
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3^{4}& 2^{2}\cdot23^{2}& 13^{3}& 2197& 1794& 1.027\\
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1& 7^{2}\cdot47&2^{8}\cdot3^{2}& 2304& 1974&
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1.0204\\
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5^{3}& 3^{7}&2^{3}\cdot17^{2}& 2312& 510&
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1.2424\\
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1&2^{5}\cdot3\cdot5^{2}& 7^{4}& 2401& 210& 1.4557\\
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5^{2}&2^{3}\cdot3^{3}\cdot11& 7^{4}& 2401& 2310& 1.005\\
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2^{6}\cdot3& 47^{2}& 7^{4}& 2401& 1974& 1.0258\\
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2^{10}& 3^{4}\cdot17& 7^{4}& 2401& 714& 1.1846\\
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3^{2}\cdot11& 7^{4}&2^{2}\cdot5^{4}& 2500& 2310&
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1.0102\\
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5^{4}& 2^{11}&3^{5}\cdot11& 2673& 330& 1.3607\\
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1&2^{4}\cdot3^{3}\cdot7&5^{2}\cdot11^{2}& 3025& 2310&
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1.0348\\
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5^{3}\cdot7& 13^{3}&2^{10}\cdot3& 3072& 2730& 1.0149\\
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53& 2^{10}\cdot3& 5^{5}& 3125& 1590& 1.0917\\
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2^{7}& 3^{4}\cdot37& 5^{5}& 3125& 1110& 1.1476\\
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11& 5^{5}&2^{6}\cdot7^{2}& 3136& 770&
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1.2113\\
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2^{10}& 3^{7}&13^{2}\cdot19& 3211& 1482&
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1.1059\\
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5^{2}& 2^{7}\cdot3^{3}& 59^{2}& 3481& 1770& 1.0904\\
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3^{3}\cdot17& 5^{5}&2^{9}\cdot7& 3584& 3570& 1.0005\\
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2^{7}& 3^{6}\cdot5&7^{3}\cdot11& 3773& 2310& 1.0633\\
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2^{3}\cdot3^{4}& 5^{5}&7^{3}\cdot11& 3773& 2310& 1.0633\\
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1& 13^{2}\cdot23&2^{4}\cdot3^{5}& 3888& 1794&
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1.1032\\
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1& 2^{7}\cdot31&3^{4}\cdot7^{2}& 3969& 1302&
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1.1554\\
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2^{9}& 59^{2}&3\cdot11^{3}& 3993& 3894& 1.003\\
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7& 3\cdot11^{3}&2^{5}\cdot5^{3}& 4000& 2310&
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1.0709\\
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1&3^{2}\cdot5\cdot7\cdot13& 2^{12}& 4096& 2730& 1.0513\\
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5^{3}& 11\cdot19^{2}& 2^{12}& 4096& 2090& 1.088\\
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3\cdot5^{3}& 61^{2}& 2^{12}& 4096& 1830& 1.1073\\
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11& 2^{12}&3\cdot37^{2}& 4107& 2442& 1.0666\\
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5\cdot7& 2^{12}&3^{5}\cdot17& 4131& 3570& 1.0178\\
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2^{8}& 3^{4}\cdot7^{2}&5^{2}\cdot13^{2}& 4225& 2730&
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1.0552\\
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2^{11}& 3^{7}&5\cdot7\cdot11^{2}& 4235& 2310&
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1.0783\\
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1& 2\cdot3^{7}&5^{4}\cdot7& 4375& 210& 1.5679\\
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7^{2}& 2^{8}\cdot19& 17^{3}& 4913& 4522& 1.0099\\
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2^{7}& 17^{3}& 71^{2}& 5041& 2414& 1.0945\\
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\end{array}$$
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\newpage
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$$\begin{array}{llllll}
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a & b & c & c & \!\!\!\mbox{\rm cond}(\mbox{\rm rad}) &
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\displaystyle \frac{\log(c)}{\log(\mbox{\rm cond)}}\\\hline
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17& 3^{6}\cdot7&2^{10}\cdot5& 5120& 3570& 1.0441\\
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3^{7}& 5^{5}&2^{6}\cdot83& 5312& 2490& 1.0969\\
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11^{3}& 2^{12}&3^{4}\cdot67& 5427& 4422& 1.0244\\
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7& 3^{2}\cdot5^{4}&2^{9}\cdot11& 5632& 2310& 1.1151\\
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7^{4}& 3^{3}\cdot5^{3}&2^{4}\cdot19^{2}& 5776& 3990&
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1.0446\\
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1& 7^{3}\cdot17&2^{3}\cdot3^{6}& 5832& 714&
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1.3196\\
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19& 5^{3}\cdot7^{2}&2^{11}\cdot3& 6144& 3990& 1.0521\\
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3^{2}& 79^{2}&2\cdot5^{5}& 6250& 2370& 1.1248\\
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1& 3^{4}\cdot79&2^{8}\cdot5^{2}& 6400& 2370&
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1.1278\\
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1& 2^{5}\cdot5\cdot41& 3^{8}& 6561& 1230& 1.2353\\
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7\cdot23& 2^{8}\cdot5^{2}& 3^{8}& 6561& 4830& 1.0361\\
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17^{2}& 2^{7}\cdot7^{2}& 3^{8}& 6561& 714& 1.3376\\
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2^{6}\cdot5& 79^{2}& 3^{8}& 6561& 2370& 1.131\\
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31^{2}&2^{5}\cdot5^{2}\cdot7& 3^{8}& 6561& 6510& 1.0009\\
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2^{10}& 7^{2}\cdot113& 3^{8}& 6561& 4746& 1.0383\\
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47^{2}& 2^{8}\cdot17& 3^{8}& 6561& 4794& 1.037\\
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7^{4}& 2^{6}\cdot5\cdot13& 3^{8}& 6561& 2730& 1.1108\\
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2^{6}& 3^{8}&5^{3}\cdot53& 6625& 1590& 1.1936\\
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2^{6}& 3\cdot13^{3}&5\cdot11^{3}& 6655& 4290& 1.0525\\
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1& 5\cdot11^{3}&2^{9}\cdot13& 6656& 1430& 1.2117\\
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2^{4}\cdot7^{2}& 3^{5}\cdot5^{2}& 19^{3}& 6859& 3990& 1.0653\\
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1& 19^{3}&2^{2}\cdot5\cdot7^{3}& 6860& 1330&
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1.2281\\
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2^{4}& 19^{3}&5^{4}\cdot11& 6875& 2090& 1.1558\\
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53& 19^{3}&2^{8}\cdot3^{3}& 6912& 6042&
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1.0155\\
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7\cdot13^{2}& 3^{8}&2^{6}\cdot11^{2}& 7744& 6006&
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1.0292\\
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3^{2}\cdot7^{3}& 17^{3}&2^{6}\cdot5^{3}& 8000& 3570&
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1.0986\\
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19& 2^{6}\cdot5^{3}&3^{6}\cdot11& 8019& 6270& 1.0281\\
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3^{2}& 7^{2}\cdot167& 2^{13}& 8192& 7014& 1.0175\\
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11& 3^{4}\cdot101& 2^{13}& 8192& 6666& 1.0234\\
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1& 5^{2}\cdot7^{3}&2^{7}\cdot67& 8576& 4690& 1.0714\\
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7^{3}& 5\cdot41^{2}&2^{2}\cdot3^{7}& 8748& 8610&
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1.0018\\
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2^{4}& 5\cdot43^{2}&3^{3}\cdot7^{3}& 9261& 9030&
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1.0028\\
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3^{2}\cdot5^{3}& 2^{13}&7\cdot11^{3}& 9317& 2310& 1.1801\\
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7\cdot13^{2}& 2^{13}&3\cdot5^{5}& 9375& 2730& 1.1559\\
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1& 3\cdot5^{5}&2^{5}\cdot293& 9376& 8790&
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1.0071\\
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1&2^{6}\cdot3\cdot7^{2}& 97^{2}& 9409& 4074& 1.1007\\
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1&2^{3}\cdot5^{2}\cdot7^{2}&3^{4}\cdot11^{2}& 9801& 2310&
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1.1866\\
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5^{2}& 3^{4}\cdot11^{2}&2\cdot17^{3}& 9826& 5610& 1.0649\\
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5^{5}& 19^{3}&2^{8}\cdot3\cdot13& 9984& 7410&
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1.0335\\
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13^{3}& 3^{3}\cdot17^{2}&2^{4}\cdot5^{4}& 10000& 6630&
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1.0467\\
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\end{array}$$
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\newpage
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The table was created using the following simple {\tt Magma} program.
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\begin{verbatim}
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// abc.m
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intrinsic Radical(N::RngIntElt) -> RngIntElt
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{Returns the product of the primes dividing N.}
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if N eq 0 then
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return 0 ;
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end if;
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return &*[x[1] : x in Factorization(N)];
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end intrinsic;
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function abc_help(c)
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ans := [];
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for a in [1..Integers()!Round(c/2)+1] do
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b := c - a;
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if Gcd([a,b,c]) eq 1 and
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c ge Radical(a*b*c) then
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Append(~ans, [a,b]);
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end if;
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end for;
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return ans;
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end function;
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intrinsic abc(start::RngIntElt, stop::RngIntElt) -> SeqEnum
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{Returns the solutions a+b = c with a<b, gcd([a,b,c])=1,
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and Radical(a*b*c) <= c.}
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return &cat[abc_help(c) : c in [start..stop]];
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end intrinsic;
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intrinsic abc_embelish(abclist::SeqEnum) -> SeqEnum
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{Given the output of abc, add in the additional
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information of c, rad(a*b*c), and log(c)/log(rad(a*b*c)).}
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return
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[<s[1], s[2], s[1]+s[2], Radical(s[1]*s[2]*(s[1]+s[2])),
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Log(s[1]+s[2]) / Log(Radical(s[1]*s[2]*(s[1]+s[2])))> : s in abclist];
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end intrinsic;
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\end{verbatim}
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\end{document}
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