20191106, 09:39  #34  
Mar 2015
Australia
2·41 Posts 
Thanks Ed, I really hope the new pairs are useful, and would love to hear how they are useful!
In the future I'd like to post the pairs list more often, but without creating a new post each time. Is there a way to create a sticky post I can update, or a link to the file I can update? The file with each pair factored takes more work so will be less often. Andrew Quote:


20191113, 09:32  #35 
Oct 2011
2^{4}·3·7 Posts 
Andrew,
Paul Zimmermann's program is running on my computer. He found a cycle 2 that is not in your list of 910 pairs. 0 217294*I  668517 = [[18*I + 23, 1], [6*I + 11, 1], [2*I + 1, 2], [2*I + 3, 1], [93*I + 52, 1]] Factors of N(z) = 494131661725 = 5^2 * 13 * 157 * 853 * 11353 1 882506*I + 263517 = [[120*I + 29, 1], [6*I + 1, 1], [2*I + 1, 2], [12*I + 245, 1]] Factors of N(z) = 848258049325 = 5^2 * 37 * 15241 * 60169 JeanLuc 
20191114, 10:10  #36 
Mar 2015
Australia
122_{8} Posts 
Thanks Jean, confirm it's a new pair will add it to my lists!
This looks like it is just outside the range of one of my searches for (1+2*I)^2 *4 factors. Will try to rediscover it! Andrew 
20191120, 10:05  #37 
Oct 2011
2^{4}×3×7 Posts 
Andrew,
Paul Zimmermann found with his own computer, two new cycles 2 that are not in your list of 0 637121*I  388928 = [[53*I + 28, 1], [12*I + 23, 1], [22*I + 15, 1], [3*I + 2, 1], [2*I + 1, 2]] Factors of N(z) = 557188157825 = 5^2 * 13 * 673 * 709 * 3593 1 455119*I + 708608 = [[4*I + 1, 1], [2*I + 1, 2], [20*I + 27, 1], [1215*I + 44, 1]] Factors of N(z) = 709258601825 = 5^2 * 17 * 1129 * 1478161 0 13357*I + 942574 = [[24*I + 1, 1], [3*I + 2, 1], [2*I + 1, 2], [2*I + 15, 1], [142*I + 23, 1]] Factors of N(z) = 888624154925 = 5^2 * 13 * 229 * 577 * 20693 1 966683*I + 116906 = [[330*I + 29, 1], [4*I + 1, 1], [2*I + 1, 2], [5*I + 2, 1], [5*I + 26, 1]] Factors of N(z) = 948143035325 = 5^2 * 17 * 29 * 701 * 109741 JeanLuc 
20191124, 18:23  #38 
Oct 2011
2^{4}×3×7 Posts 
Paul Zimmermann found 10 new complex 2cycles with his program on his own computer :
0 916649*I  885682 Factors of z taken in the first quadrant : [[288*I + 215, 1], [2*I + 1, 2], [5*I + 8, 1], [18*I + 73, 1]] Factors of N(z) = 1624677994325 = 5^2 * 89 * 5653 * 129169 1 463049*I  770318 Factors of z taken in the first quadrant : [[3*I + 8, 1], [3*I + 2, 1], [2*I + 1, 2], [4*I + 9, 1], [2*I + 13, 1], [2*I + 45, 1]] Factors of N(z) = 807804197525 = 5^2 * 13 * 73 * 97 * 173 * 2029 0 20345*I  864140 Factors of z taken in the first quadrant : [[6*I + 5, 1], [I + 2, 1], [2*I + 1, 2], [34*I + 21, 1], [29*I + 246, 1]] Factors of N(z) = 747151858625 = 5^3 * 61 * 1597 * 61357 1 1216457*I  263476 Factors of z taken in the first quadrant : [[334*I + 711, 1], [4*I + 11, 1], [2*I + 1, 2], [27*I + 2, 1]] Factors of N(z) = 1549187235425 = 5^2 * 137 * 733 * 617077 0 803372*I  855881 Factors of z taken in the first quadrant : [[14*I + 1, 1], [2*I + 1, 1], [35924*I + 10411, 1]] Factors of N(z) = 1377938856545 = 5 * 197 * 1398922697 1 274380*I  543575 Factors of z taken in the first quadrant : [[94*I + 109, 1], [19*I + 46, 1], [4*I + 1, 1], [I + 2, 1], [2*I + 1, 1], [I + 4, 1]] Factors of N(z) = 370758165025 = 5^2 * 17^2 * 2477 * 20717 0 277267*I  779326 Factors of z taken in the first quadrant : [[10*I + 29, 1], [4*I + 1, 1], [3*I + 2, 1], [2*I + 1, 1], [3*I + 10, 1], [66*I + 41, 1]] Factors of N(z) = 684226003565 = 5 * 13 * 17 * 109 * 941 * 6037 1 1090733*I + 131326 Factors of z taken in the first quadrant : [[4*I + 1, 1], [2*I + 1, 1], [45*I + 8, 1], [2200*I + 1399, 1]] Factors of N(z) = 1206944995565 = 5 * 17 * 2089 * 6797201 0 941792*I  771944 Factors of z taken in the first quadrant : [[775*I + 174, 1], [8*I + 7, 1], [3*I + 2, 1], [I + 1, 6], [2*I + 1, 2]] Factors of N(z) = 1482869710400 = 2^6 * 5^2 * 13 * 113 * 630901 1 1003408*I  743656 Factors of z taken in the first quadrant : [[930*I + 209, 1], [6*I + 1, 1], [I + 1, 6], [2*I + 1, 2], [2*I + 5, 1]] Factors of N(z) = 1559851860800 = 2^6 * 5^2 * 29 * 37 * 908581 0 461397*I  629476 Factors of z taken in the first quadrant : [[696*I + 431, 1], [6*I + 1, 1], [4*I + 1, 1], [2*I + 1, 1], [I + 4, 2]] Factors of N(z) = 609127226185 = 5 * 17^3 * 37 * 670177 1 963243*I + 439396 Factors of z taken in the first quadrant : [[6*I + 1, 1], [2*I + 1, 1], [15*I + 2, 1], [44*I + 31, 1], [22*I + 93, 1]] Factors of N(z) = 1120905921865 = 5 * 37 * 229 * 2897 * 9133 0 1194502*I  594811 Factors of z taken in the first quadrant : [[424*I + 581, 1], [6*I + 1, 1], [6*I + 5, 1], [2*I + 1, 2], [5*I + 6, 1]] Factors of N(z) = 1780635153725 = 5^2 * 37 * 61^2 * 517337 1 732458*I + 1207531 Factors of z taken in the first quadrant : [[3*I + 38, 1], [6*I + 1, 1], [2*I + 1, 2], [5*I + 6, 1], [152*I + 35, 1]] Factors of N(z) = 1994625837725 = 5^2 * 37 * 61 * 1453 * 24329 0 947796*I + 166087 Factors of z taken in the first quadrant : [[44*I + 61, 1], [4*I + 1, 2], [2*I + 1, 1], [2*I + 3, 1], [93*I + 8, 1]] Factors of N(z) = 925902149185 = 5 * 13 * 17^2 * 5657 * 8713 1 121716*I + 1192073 Factors of z taken in the first quadrant : [[2028*I + 965, 1], [14*I + 1, 1], [4*I + 1, 2], [2*I + 1, 1]] Factors of N(z) = 1435852821985 = 5 * 17^2 * 197 * 5044009 0 1175036*I + 248148 Factors of z taken in the first quadrant : [[780*I + 259, 1], [6*I + 11, 1], [4*I + 1, 1], [I + 1, 5], [2*I + 1, 2]] Factors of N(z) = 1442287031200 = 2^5 * 5^2 * 17 * 157 * 675481 1 1149364*I + 422652 Factors of z taken in the first quadrant : [[46*I + 21, 1], [I + 1, 5], [2*I + 1, 2], [4*I + 5, 1], [120*I + 59, 1]] Factors of N(z) = 1499672317600 = 2^5 * 5^2 * 41 * 2557 * 17881 0 1043992*I + 514219 Factors of z taken in the first quadrant : [[30*I + 161, 1], [15*I + 4, 1], [4*I + 1, 2], [2*I + 1, 2], [2*I + 5, 1]] Factors of N(z) = 1354340476025 = 5^2 * 17^2 * 29 * 241 * 26821 1 358192*I + 1120781 Factors of z taken in the first quadrant : [[2*I + 1, 2], [2*I + 5, 1], [7*I + 2, 1], [25*I + 4, 1], [228*I + 65, 1]] Factors of N(z) = 1384451558825 = 5^2 * 29 * 53 * 641 * 56209 
20191125, 07:14  #39 
Mar 2015
Australia
2×41 Posts 
Thanks for the updates! I'll add these to my list, have also found a number
of others since my last update. I suspect most of those Paul found and I missed are due to the factors being beyond the limits of my current search, so it's good to have a search very different from mine! Please keep an eye out for any which a) Don't have 1 or I as the unit factor; or b) don't have 1+I, 1+2*I or 1+ 4*I as one of the factors I suspect b) will be found eventually (similar to regular amicable pairs not divisible by 6 or even scarcer not divisible by 30) a) I suspect is true, maybe there is a simple proof? Andrew 
20191130, 04:00  #40 
Mar 2015
Australia
82_{10} Posts 
Thanks again for the updates Paul and JeanLuc! Have just updated my list with your pairs and a few new ones of mine which takes it to 943 pairs. Have made this 7a, 8 will be at 1000 or more!
As always, any errors, omissions or other problems please let me know! Andrew 
20191130, 11:14  #41 
Oct 2011
2^{4}×3×7 Posts 
OK, a lot of thanks Andrew.

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