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