Open in CoCalc
1\documentclass{article}
2\centerline{$c\leq 10000$}
3\begin{document}
4$$\begin{array}{llllll} 5 a & b & c & c & \!\!\!\mbox{\rm cond}(\mbox{\rm rad}) & 6 \displaystyle \frac{\log(c)}{\log(\mbox{\rm cond)}}\\\hline 7 1& 1& 2& 2& 2& 1\\ 8 1& 2^{3}& 3^{2}& 9& 6& 1.2263\\ 9 5& 3^{3}& 2^{5}& 32& 30& 1.019\\ 10 1& 2^{4}\cdot3& 7^{2}& 49& 42& 1.0412\\ 11 1& 3^{2}\cdot7& 2^{6}& 64& 42& 1.1127\\ 12 1& 2^{4}\cdot5& 3^{4}& 81& 30& 1.292\\ 13 2^{5}& 7^{2}& 3^{4}& 81& 42& 1.1757\\ 14 2^{2}& 11^{2}& 5^{3}& 125& 110& 1.0272\\ 15 3& 5^{3}& 2^{7}& 128& 30& 1.4266\\ 16 1& 2^{5}\cdot7&3^{2}\cdot5^{2}& 225& 210&1.0129\\ 17 1& 2\cdot11^{2}& 3^{5}& 243& 66& 1.3111\\ 18 2& 3^{5}&5\cdot7^{2}& 245& 210& 1.0288\\ 19 7& 3^{5}&2\cdot5^{3}& 250& 210& 1.0326\\ 20 13& 3^{5}& 2^{8}& 256& 78& 1.2728\\ 21 3^{4}& 5^{2}\cdot7& 2^{8}& 256& 210& 1.037\\ 22 1& 2^{5}\cdot3^{2}& 17^{2}& 289& 102& 1.2252\\ 23 2^{2}\cdot5^{2}& 3^{5}& 7^{3}& 343& 210& 1.0918\\ 24 2^{5}& 7^{3}&3\cdot5^{3}& 375& 210& 1.1084\\ 25 5& 3\cdot13^{2}& 2^{9}& 512& 390& 1.0456\\ 26 13^{2}& 7^{3}& 2^{9}& 512& 182& 1.1988\\ 27 1& 2^{9}&3^{3}\cdot19& 513& 114& 1.3176\\ 28 3^{3}& 2^{9}&7^{2}\cdot11& 539& 462& 1.0251\\ 29 1& 2^{4}\cdot3\cdot13& 5^{4}& 625& 390& 1.079\\ 30 7^{2}& 2^{6}\cdot3^{2}& 5^{4}& 625& 210& 1.204\\ 31 3^{4}& 2^{5}\cdot17& 5^{4}& 625& 510& 1.0326\\ 32 1& 3^{3}\cdot5^{2}&2^{2}\cdot13^{2}& 676& 390& 1.0922\\ 33 1& 2^{3}\cdot7\cdot13& 3^{6}& 729& 546& 1.0459\\ 34 5^{2}& 2^{6}\cdot11& 3^{6}& 729& 330& 1.1367\\ 35 2^{3}\cdot13& 5^{4}& 3^{6}& 729& 390& 1.1048\\ 36 2^{3}\cdot5^{2}& 23^{2}& 3^{6}& 729& 690& 1.0084\\ 37 1& 2^{6}\cdot3\cdot5& 31^{2}& 961& 930& 1.0048\\ 38 7^{3}& 5^{4}&2^{3}\cdot11^{2}& 968& 770& 1.0344\\ 39 1& 2^{10}&5^{2}\cdot41& 1025& 410& 1.1523\\ 40 5& 2^{10}&3\cdot7^{3}& 1029& 210& 1.2972\\ 41 1& 3^{5}\cdot5&2^{6}\cdot19& 1216& 570& 1.1194\\ 42 2^{3}& 3^{3}\cdot7^{2}& 11^{3}& 1331& 462& 1.1725\\ 43 3^{4}& 2\cdot5^{4}& 11^{3}& 1331& 330& 1.2405\\ 44 3^{5}& 2^{6}\cdot17& 11^{3}& 1331& 1122& 1.0243\\ 45 2^{7}\cdot5& 3^{6}& 37^{2}& 1369& 1110& 1.0299\\ 46 2^{8}& 11^{3}&3\cdot23^{2}& 1587& 1518& 1.0061\\ 47 3^{4}& 2^{6}\cdot5^{2}& 41^{2}& 1681& 1230& 1.0439\\ 48 49 50\end{array} 51$$
52
53\newpage
54$$\begin{array}{llllll} 55 a & b & c & c & \!\!\!\mbox{\rm cond}(\mbox{\rm rad}) & 56 \displaystyle \frac{\log(c)}{\log(\mbox{\rm cond)}}\\\hline 57 23& 3^{4}\cdot5^{2}& 2^{11}& 2048& 690& 1.1664\\ 58 5^{2}& 7\cdot17^{2}& 2^{11}& 2048& 1190& 1.0767\\ 59 3^{5}& 5\cdot19^{2}& 2^{11}& 2048& 570& 1.2016\\ 60 3^{2}& 2^{11}&11^{2}\cdot17& 2057& 1122& 611.0863\\ 62 11& 2^{7}\cdot17& 3^{7}& 2187& 1122& 1.095\\ 63 139& 2^{11}& 3^{7}& 2187& 834& 1.1433\\ 64 2^{9}& 5^{2}\cdot67& 3^{7}& 2187& 2010& 1.0111\\ 65 2\cdot5& 3^{7}& 13^{3}& 2197& 390& 1.2898\\ 66 3^{4}& 2^{2}\cdot23^{2}& 13^{3}& 2197& 1794& 1.027\\ 67 1& 7^{2}\cdot47&2^{8}\cdot3^{2}& 2304& 1974& 681.0204\\ 69 5^{3}& 3^{7}&2^{3}\cdot17^{2}& 2312& 510& 701.2424\\ 71 1&2^{5}\cdot3\cdot5^{2}& 7^{4}& 2401& 210& 1.4557\\ 72 5^{2}&2^{3}\cdot3^{3}\cdot11& 7^{4}& 2401& 2310& 1.005\\ 73 2^{6}\cdot3& 47^{2}& 7^{4}& 2401& 1974& 1.0258\\ 74 2^{10}& 3^{4}\cdot17& 7^{4}& 2401& 714& 1.1846\\ 75 3^{2}\cdot11& 7^{4}&2^{2}\cdot5^{4}& 2500& 2310& 761.0102\\ 77 5^{4}& 2^{11}&3^{5}\cdot11& 2673& 330& 1.3607\\ 78 1&2^{4}\cdot3^{3}\cdot7&5^{2}\cdot11^{2}& 3025& 2310& 791.0348\\ 80 5^{3}\cdot7& 13^{3}&2^{10}\cdot3& 3072& 2730& 1.0149\\ 81 53& 2^{10}\cdot3& 5^{5}& 3125& 1590& 1.0917\\ 82 2^{7}& 3^{4}\cdot37& 5^{5}& 3125& 1110& 1.1476\\ 83 11& 5^{5}&2^{6}\cdot7^{2}& 3136& 770& 841.2113\\ 85 2^{10}& 3^{7}&13^{2}\cdot19& 3211& 1482& 861.1059\\ 87 5^{2}& 2^{7}\cdot3^{3}& 59^{2}& 3481& 1770& 1.0904\\ 88 3^{3}\cdot17& 5^{5}&2^{9}\cdot7& 3584& 3570& 1.0005\\ 89 2^{7}& 3^{6}\cdot5&7^{3}\cdot11& 3773& 2310& 1.0633\\ 90 2^{3}\cdot3^{4}& 5^{5}&7^{3}\cdot11& 3773& 2310& 1.0633\\ 91 1& 13^{2}\cdot23&2^{4}\cdot3^{5}& 3888& 1794& 921.1032\\ 93 1& 2^{7}\cdot31&3^{4}\cdot7^{2}& 3969& 1302& 941.1554\\ 95 2^{9}& 59^{2}&3\cdot11^{3}& 3993& 3894& 1.003\\ 96 7& 3\cdot11^{3}&2^{5}\cdot5^{3}& 4000& 2310& 971.0709\\ 98 1&3^{2}\cdot5\cdot7\cdot13& 2^{12}& 4096& 2730& 1.0513\\ 99 5^{3}& 11\cdot19^{2}& 2^{12}& 4096& 2090& 1.088\\ 100 3\cdot5^{3}& 61^{2}& 2^{12}& 4096& 1830& 1.1073\\ 101 11& 2^{12}&3\cdot37^{2}& 4107& 2442& 1.0666\\ 102 5\cdot7& 2^{12}&3^{5}\cdot17& 4131& 3570& 1.0178\\ 103 2^{8}& 3^{4}\cdot7^{2}&5^{2}\cdot13^{2}& 4225& 2730& 1041.0552\\ 105 2^{11}& 3^{7}&5\cdot7\cdot11^{2}& 4235& 2310& 1061.0783\\ 107 1& 2\cdot3^{7}&5^{4}\cdot7& 4375& 210& 1.5679\\ 108 7^{2}& 2^{8}\cdot19& 17^{3}& 4913& 4522& 1.0099\\ 109 2^{7}& 17^{3}& 71^{2}& 5041& 2414& 1.0945\\ 110\end{array}$$
111
112\newpage
113$$\begin{array}{llllll} 114 a & b & c & c & \!\!\!\mbox{\rm cond}(\mbox{\rm rad}) & 115 \displaystyle \frac{\log(c)}{\log(\mbox{\rm cond)}}\\\hline 116 17& 3^{6}\cdot7&2^{10}\cdot5& 5120& 3570& 1.0441\\ 117 3^{7}& 5^{5}&2^{6}\cdot83& 5312& 2490& 1.0969\\ 118 11^{3}& 2^{12}&3^{4}\cdot67& 5427& 4422& 1.0244\\ 119 7& 3^{2}\cdot5^{4}&2^{9}\cdot11& 5632& 2310& 1.1151\\ 120 7^{4}& 3^{3}\cdot5^{3}&2^{4}\cdot19^{2}& 5776& 3990& 1211.0446\\ 122 1& 7^{3}\cdot17&2^{3}\cdot3^{6}& 5832& 714& 1231.3196\\ 124 19& 5^{3}\cdot7^{2}&2^{11}\cdot3& 6144& 3990& 1.0521\\ 125 3^{2}& 79^{2}&2\cdot5^{5}& 6250& 2370& 1.1248\\ 126 1& 3^{4}\cdot79&2^{8}\cdot5^{2}& 6400& 2370& 1271.1278\\ 128 1& 2^{5}\cdot5\cdot41& 3^{8}& 6561& 1230& 1.2353\\ 129 7\cdot23& 2^{8}\cdot5^{2}& 3^{8}& 6561& 4830& 1.0361\\ 130 17^{2}& 2^{7}\cdot7^{2}& 3^{8}& 6561& 714& 1.3376\\ 131 2^{6}\cdot5& 79^{2}& 3^{8}& 6561& 2370& 1.131\\ 132 31^{2}&2^{5}\cdot5^{2}\cdot7& 3^{8}& 6561& 6510& 1.0009\\ 133 2^{10}& 7^{2}\cdot113& 3^{8}& 6561& 4746& 1.0383\\ 134 47^{2}& 2^{8}\cdot17& 3^{8}& 6561& 4794& 1.037\\ 135 7^{4}& 2^{6}\cdot5\cdot13& 3^{8}& 6561& 2730& 1.1108\\ 136 2^{6}& 3^{8}&5^{3}\cdot53& 6625& 1590& 1.1936\\ 137 2^{6}& 3\cdot13^{3}&5\cdot11^{3}& 6655& 4290& 1.0525\\ 138 1& 5\cdot11^{3}&2^{9}\cdot13& 6656& 1430& 1.2117\\ 139 2^{4}\cdot7^{2}& 3^{5}\cdot5^{2}& 19^{3}& 6859& 3990& 1.0653\\ 140 1& 19^{3}&2^{2}\cdot5\cdot7^{3}& 6860& 1330& 1411.2281\\ 142 2^{4}& 19^{3}&5^{4}\cdot11& 6875& 2090& 1.1558\\ 143 53& 19^{3}&2^{8}\cdot3^{3}& 6912& 6042& 1441.0155\\ 145 7\cdot13^{2}& 3^{8}&2^{6}\cdot11^{2}& 7744& 6006& 1461.0292\\ 147 3^{2}\cdot7^{3}& 17^{3}&2^{6}\cdot5^{3}& 8000& 3570& 1481.0986\\ 149 19& 2^{6}\cdot5^{3}&3^{6}\cdot11& 8019& 6270& 1.0281\\ 150 3^{2}& 7^{2}\cdot167& 2^{13}& 8192& 7014& 1.0175\\ 151 11& 3^{4}\cdot101& 2^{13}& 8192& 6666& 1.0234\\ 152 1& 5^{2}\cdot7^{3}&2^{7}\cdot67& 8576& 4690& 1.0714\\ 153 7^{3}& 5\cdot41^{2}&2^{2}\cdot3^{7}& 8748& 8610& 1541.0018\\ 155 2^{4}& 5\cdot43^{2}&3^{3}\cdot7^{3}& 9261& 9030& 1561.0028\\ 157 3^{2}\cdot5^{3}& 2^{13}&7\cdot11^{3}& 9317& 2310& 1.1801\\ 158 7\cdot13^{2}& 2^{13}&3\cdot5^{5}& 9375& 2730& 1.1559\\ 159 1& 3\cdot5^{5}&2^{5}\cdot293& 9376& 8790& 1601.0071\\ 161 1&2^{6}\cdot3\cdot7^{2}& 97^{2}& 9409& 4074& 1.1007\\ 162 1&2^{3}\cdot5^{2}\cdot7^{2}&3^{4}\cdot11^{2}& 9801& 2310& 1631.1866\\ 164 5^{2}& 3^{4}\cdot11^{2}&2\cdot17^{3}& 9826& 5610& 1.0649\\ 165 5^{5}& 19^{3}&2^{8}\cdot3\cdot13& 9984& 7410& 1661.0335\\ 167 13^{3}& 3^{3}\cdot17^{2}&2^{4}\cdot5^{4}& 10000& 6630& 1681.0467\\ 169\end{array}$$
170
171\newpage
172The table was created using the following simple {\tt Magma} program.
173\begin{verbatim}
174// abc.m
175
177{Returns the product of the primes dividing N.}
178   if N eq 0 then
179      return 0 ;
180   end if;
181   return &*[x[1] : x in Factorization(N)];
182end intrinsic;
183
184function abc_help(c)
185   ans := [];
186   for a in [1..Integers()!Round(c/2)+1] do
187      b := c - a;
188      if Gcd([a,b,c]) eq 1 and
190         Append(~ans, [a,b]);
191      end if;
192   end for;
193   return ans;
194end function;
195
196intrinsic abc(start::RngIntElt, stop::RngIntElt) -> SeqEnum
197{Returns the solutions a+b = c with a<b, gcd([a,b,c])=1,
199   return &cat[abc_help(c) : c in [start..stop]];
200end intrinsic;
201
202intrinsic abc_embelish(abclist::SeqEnum) -> SeqEnum