Forum de mathématiques - Bibm@th.net

Bonjour Egaciole,

"De manière générale, considérons 3x + b1 et 3x + b2, les cycles de 3x + b1 et de 3x + b2 se retrouvent dans 3x + (b1 * b2). Je pense que c'est assez facile à démontrer d'ailleurs."

Puisque c'est assez facile à démontrer, je veux bien en voir une démonstration, car cette assertion est d'évidence fausse.
Par contre si b1 est de la forme 3^m alors les cycles b1 * b2 seront ceux de b2 multipliés par 3^m.

Concernant les cycles 3x+d, d>0, (Kaneda, 2014, [62]) : Pour tout d ∈ Z avec d ≡ 1(mod 6) ou d ≡ −1 (mod 6), le nombre de (3x + d)-cycles est fini.

Je prétend que les seuls valeurs de d (d>0)  donnant un seul cycle sont de la forme 3^m (m>=0)
La liste ci-dessous donne le nombre de cycles et le début de chaque cycle pour 1<=d<1000
d=1,c:1{1}
d=3,c:1{3}
d=5,c:6{1,5,19,23,187,347}
d=7,c:2{5,7}
d=9,c:1{9}
d=11,c:3{1,11,13}
d=13,c:10{1,13,131,211,227,251,259,283,287,319}
d=15,c:6{3,15,57,69,561,1041}
d=17,c:3{1,17,23}
d=19,c:2{5,19}
d=21,c:2{15,21}
d=23,c:4{5,7,23,41}
d=25,c:8{5,7,17,25,95,115,935,1735}
d=27,c:1{27}
d=29,c:5{1,11,29,3811,7055}
d=31,c:2{13,31}
d=33,c:3{3,33,39}
d=35,c:9{7,13,17,25,35,133,161,1309,2429}
d=37,c:4{19,23,29,37}
d=39,c:10{3,39,393,633,681,753,777,849,861,957}
d=41,c:2{1,41}
d=43,c:2{1,43}
d=45,c:6{9,45,171,207,1683,3123}
d=47,c:8{5,25,47,65,73,85,89,101}
d=49,c:3{25,35,49}
d=51,c:3{3,51,69}
d=53,c:2{53,103}
d=55,c:11{1,5,7,11,41,55,65,209,253,2057,3817}
d=57,c:2{15,57}
d=59,c:8{1,59,133,149,181,185,217,221}
d=61,c:3{1,61,235}
d=63,c:2{45,63}
d=65,c:16{5,13,19,65,247,299,655,1055,1135,1255,1295,1415,1435,1595,2431,4511}
d=67,c:2{17,67}
d=69,c:4{15,21,69,123}
d=71,c:8{29,31,71,2585,2809,3985,4121,4409}
d=73,c:4{5,19,47,73}
d=75,c:8{15,21,51,75,285,345,2805,5205}
d=77,c:5{1,7,55,77,91}
d=79,c:5{1,7,79,233,265}
d=81,c:1{81}
d=83,c:4{65,83,109,157}
d=85,c:9{5,7,17,85,115,323,391,3179,5899}
d=87,c:5{3,33,87,11433,21165}
d=89,c:2{17,89}
d=91,c:14{1,7,25,59,65,91,917,1477,1589,1757,1813,1981,2009,2233}
d=93,c:2{39,93}
d=95,c:10{1,17,19,23,25,95,361,437,3553,6593}
d=97,c:3{1,13,97}
d=99,c:3{9,99,117}
d=101,c:8{7,11,19,23,29,31,37,101}
d=103,c:3{5,23,103}
d=105,c:9{21,39,51,75,105,399,483,3927,7287}
d=107,c:2{1,107}
d=109,c:2{19,109}
d=111,c:4{57,69,87,111}
d=113,c:2{1,113}
d=115,c:11{13,17,23,25,35,115,205,437,529,4301,7981}
d=117,c:10{9,117,1179,1899,2043,2259,2331,2547,2583,2871}
d=119,c:9{1,5,7,11,23,85,119,125,161}
d=121,c:5{5,11,19,121,143}
d=123,c:2{3,123}
d=125,c:12{1,25,35,47,85,125,143,475,575,899,4675,8675}
d=127,c:3{1,41,127}
d=129,c:2{3,129}
d=131,c:4{13,17,23,131}
d=133,c:5{11,35,59,95,133}
d=135,c:6{27,135,513,621,5049,9369}
d=137,c:6{1,41,137,503,743,967}
d=139,c:2{11,139}
d=141,c:8{15,75,141,195,219,255,267,303}
d=143,c:15{7,11,13,17,29,143,169,1441,2321,2497,2761,2849,3113,3157,3509}
d=145,c:13{1,5,23,29,47,55,145,551,667,5423,10063,19055,35275}
d=147,c:3{75,105,147}
d=149,c:3{19,149,667}
d=151,c:3{41,113,151}
d=153,c:3{9,153,207}
d=155,c:8{1,31,65,155,589,713,5797,10757}
d=157,c:2{13,157}
d=159,c:2{159,309}
d=161,c:9{11,19,25,35,49,79,115,161,287}
d=163,c:3{11,17,163}
d=165,c:11{3,15,21,33,123,165,195,627,759,6171,11451}
d=167,c:4{13,95,127,167}
d=169,c:12{11,13,17,169,1703,2743,2951,3263,3367,3679,3731,4147}
d=171,c:2{45,171}
d=173,c:3{7,37,173}
d=175,c:18{29,35,49,53,65,73,85,89,101,103,119,125,143,175,665,805,6545,12145}
d=177,c:8{3,177,399,447,543,555,651,663}
d=179,c:2{5,179}
d=181,c:4{11,23,55,181}
d=183,c:3{3,183,705}
d=185,c:10{1,37,95,115,145,185,703,851,6919,12839}
d=187,c:7{5,11,17,43,187,221,253}
d=189,c:2{135,189}
d=191,c:4{47,191,961,2405}
d=193,c:5{1,5,31,91,193}
d=195,c:16{15,39,57,195,741,897,1965,3165,3405,3765,3885,4245,4305,4785,7293,13533}
d=197,c:2{5,197}
d=199,c:3{13,47,199}
d=201,c:2{51,201}
d=203,c:8{1,7,43,77,145,203,26677,49385}
d=205,c:8{5,41,43,205,779,943,7667,14227}
d=207,c:4{45,63,207,369}
d=209,c:8{1,19,55,175,209,235,247,359}
d=211,c:2{37,211}
d=213,c:8{87,93,213,7755,8427,11955,12363,13227}
d=215,c:10{1,5,31,43,47,215,817,989,8041,14921}
d=217,c:5{17,59,91,155,217}
d=219,c:4{15,57,141,219}
d=221,c:13{5,13,17,221,299,2227,3587,3859,4267,4403,4811,4879,5423}
d=223,c:3{35,71,223}
d=225,c:8{45,63,153,225,855,1035,8415,15615}
d=227,c:2{5,227}
d=229,c:9{7,19,23,29,31,37,47,53,229}
d=231,c:5{3,21,165,231,273}
d=233,c:20{5,233,647,727,919,983,1015,1079,1111,1159,1175,1223,1367,1439,1535,1555,1651,1655,1663,1907}
d=235,c:16{11,17,19,25,47,125,235,325,365,425,445,505,893,1081,8789,16309}
d=237,c:5{3,21,237,699,795}
d=239,c:5{1,13,19,113,239}
d=241,c:3{7,59,241}
d=243,c:1{243}
d=245,c:12{43,49,91,119,125,167,175,245,931,1127,9163,17003}
d=247,c:17{1,5,7,11,19,65,145,247,337,2489,4009,4313,4769,4921,5377,5453,6061}
d=249,c:4{195,249,327,471}
d=251,c:3{91,251,1445}
d=253,c:9{1,13,17,23,55,77,253,299,451}
d=255,c:9{15,21,51,255,345,969,1173,9537,17697}
d=257,c:4{1,257,367,635}
d=259,c:12{29,59,67,85,107,113,121,133,161,185,203,259}
d=261,c:5{9,99,261,34299,63495}
d=263,c:4{7,17,37,263}
d=265,c:10{47,49,53,67,265,515,1007,1219,9911,18391}
d=267,c:2{51,267}
d=269,c:19{55,139,179,211,223,227,251,259,269,283,287,319,323,331,347,367,383,395,431}
d=271,c:2{17,271}
d=273,c:14{3,21,75,177,195,273,2751,4431,4767,5271,5439,5943,6027,6699}
d=275,c:14{1,5,25,35,55,77,187,205,275,325,1045,1265,10285,19085}
d=277,c:2{13,277}
d=279,c:2{117,279}
d=281,c:8{1,17,19,25,37,41,281,287}
d=283,c:4{5,53,283,821}
d=285,c:10{3,51,57,69,75,285,1083,1311,10659,19779}
d=287,c:9{5,7,31,121,169,193,205,229,287}
d=289,c:5{1,13,17,289,391}
d=291,c:3{3,39,291}
d=293,c:2{11,293}
d=295,c:30{5,59,281,295,601,665,697,737,745,761,809,817,881,889,905,925,937,977,989,1009,1033,1085,1105,1117,1121,1213,1229,1357,11033,20473}
d=297,c:3{27,297,351}
d=299,c:16{5,19,23,65,91,197,299,533,3013,4853,5221,5773,5957,6509,6601,7337}
d=301,c:5{7,19,67,215,301}
d=303,c:8{21,33,57,69,87,93,111,303}
d=305,c:11{5,7,11,13,61,305,1159,1175,1403,11407,21167}
d=307,c:8{5,17,41,137,205,293,307,353}
d=309,c:3{15,69,309}
d=311,c:3{13,29,311}
d=313,c:5{5,11,35,47,313}
d=315,c:9{63,117,153,225,315,1197,1449,11781,21861}
d=317,c:3{1,317,539}
d=319,c:8{11,29,37,121,319,377,41921,77605}
d=321,c:2{3,321}
d=323,c:5{5,19,85,323,437}
d=325,c:20{25,65,83,91,95,127,221,325,1235,1495,3275,5275,5675,6275,6475,7075,7175,7975,12155,22555}
d=327,c:2{57,327}
d=329,c:13{1,11,35,175,235,295,329,407,455,511,595,623,707}
d=331,c:3{13,31,331}
d=333,c:4{171,207,261,333}
d=335,c:10{1,17,67,73,85,335,1273,1541,12529,23249}
d=337,c:9{23,25,47,59,91,107,209,227,337}
d=339,c:2{3,339}
d=341,c:6{31,143,341,403,647,1003}
d=343,c:6{13,175,221,245,343,177337}
d=345,c:11{39,51,69,75,105,345,615,1311,1587,12903,23943}
d=347,c:3{5,7,347}
d=349,c:3{11,31,349}
d=351,c:10{27,351,3537,5697,6129,6777,6993,7641,7749,8613}
d=353,c:6{1,167,191,265,319,353}
d=355,c:33{1,7,49,71,145,155,355,1349,1633,11293,11377,12925,13181,13277,13781,14045,14077,14429,15581,15997,17009,18413,18461,19925,20605,21077,22045,22909,23821,24637,26221,27709,30449}
d=357,c:9{3,15,21,33,69,255,357,375,483}
d=359,c:12{109,113,121,145,157,169,191,209,229,263,269,359}
d=361,c:12{5,17,35,53,59,79,95,97,113,131,137,361}
d=363,c:5{15,33,57,363,429}
d=365,c:13{11,13,25,41,73,95,127,235,365,1387,1679,13651,25331}
d=367,c:2{7,367}
d=369,c:2{9,369}
d=371,c:4{25,265,371,721}
d=373,c:3{23,31,373}
d=375,c:12{3,75,105,141,255,375,429,1425,1725,2697,14025,26025}
d=377,c:16{13,19,29,53,143,377,3799,6119,6583,7279,7511,8207,8323,9251,49543,91715}
d=379,c:4{35,41,61,379}
d=381,c:3{3,123,381}
d=383,c:3{41,67,383}
d=385,c:18{5,7,17,23,35,49,77,107,143,187,275,287,385,455,1463,1771,14399,26719}
d=387,c:2{9,387}
d=389,c:4{17,37,59,389}
d=391,c:10{13,23,29,37,77,85,119,391,529,697}
d=393,c:4{39,51,69,393}
d=395,c:15{5,17,29,35,47,79,83,139,395,1165,1325,1501,1817,14773,27413}
d=397,c:2{11,397}
d=399,c:5{33,105,177,285,399}
d=401,c:2{5,401}
d=403,c:12{5,31,169,403,4061,6541,7037,7781,8029,8773,8897,9889}
d=405,c:6{81,405,1539,1863,15147,28107}
d=407,c:10{1,31,37,71,119,209,253,319,407,481}
d=409,c:4{19,23,371,409}
d=411,c:6{3,123,411,1509,2229,2901}
d=413,c:12{5,7,295,413,911,931,979,1043,1267,1295,1519,1547}
d=415,c:10{23,83,325,415,545,785,1577,1909,15521,28801}
d=417,c:2{33,417}
d=419,c:3{41,419,1045}
d=421,c:3{37,67,421}
d=423,c:8{45,225,423,585,657,765,801,909}
d=425,c:15{23,25,35,73,83,85,119,137,289,425,575,1615,1955,15895,29495}
d=427,c:5{7,67,305,427,1645}
d=429,c:15{21,33,39,51,87,429,507,4323,6963,7491,8283,8547,9339,9471,10527}
d=431,c:24{7,19,25,35,37,53,65,73,85,89,101,103,119,121,125,133,143,151,157,175,431,593,737,745}
d=433,c:5{13,31,61,433,799}
d=435,c:13{3,15,69,87,141,165,435,1653,2001,16269,30189,57165,105825}
d=437,c:8{1,5,95,115,133,437,587,779}
d=439,c:11{5,41,61,65,85,95,125,149,151,439,4513}
d=441,c:3{225,315,441}
d=443,c:6{13,19,41,59,91,443}
d=445,c:9{1,29,85,89,445,1691,2047,16643,30883}
d=447,c:3{57,447,2001}
d=449,c:2{19,449}
d=451,c:5{11,41,43,451,533}
d=453,c:3{123,339,453}
d=455,c:26{5,17,31,35,43,91,103,125,133,169,221,295,325,455,1729,2093,4585,7385,7945,8785,9065,9905,10045,11165,17017,31577}
d=457,c:5{13,49,103,287,457}
d=459,c:3{27,459,621}
d=461,c:2{7,461}
d=463,c:2{29,463}
d=465,c:8{3,93,195,465,1767,2139,17391,32271}
d=467,c:4{71,107,125,467}
d=469,c:4{31,119,335,469}
d=471,c:2{39,471}
d=473,c:5{1,11,43,473,559}
d=475,c:14{5,11,29,85,95,115,125,133,323,475,1805,2185,17765,32965}
d=477,c:2{477,927}
d=479,c:5{7,37,41,49,479}
d=481,c:26{23,37,95,113,119,127,139,155,175,215,217,233,247,281,299,353,377,481,4847,7807,8399,9287,9583,10471,10619,11803}
d=483,c:9{33,57,75,105,147,237,345,483,861}
d=485,c:23{5,7,19,23,29,31,37,47,49,53,65,83,97,101,251,259,283,299,485,1843,2231,18139,33659}
d=487,c:3{43,103,487}
d=489,c:3{33,51,489}
d=491,c:3{11,49,491}
d=493,c:8{1,17,29,187,493,667,64787,119935}
d=495,c:11{9,45,63,99,369,495,585,1881,2277,18513,34353}
d=497,c:16{13,79,95,103,113,145,203,217,265,355,497,18095,19663,27895,28847,30863}
d=499,c:53{1,23,139,499,1549,2381,4189,4513,4525,4781,5005,5281,5357,5453,5501,5581,5773,5965,6029,6085,6221,6365,6605,6853,6893,7069,7073,7085,7109,7117,7201,7261,7517,7525,7585,7789,7841,7901,8225,8513,8545,8549,8581,8737,8797,8837,9061,9085,9445,10433,10465,10817,10849}
d=501,c:4{39,285,381,501}
d=503,c:8{5,7,11,19,23,29,503,509}
d=505,c:18{35,49,55,59,79,91,95,101,113,115,145,155,185,505,1919,2323,18887,35047}
d=507,c:12{33,39,51,507,5109,8229,8853,9789,10101,11037,11193,12441}
d=509,c:3{1,5,509}
d=511,c:10{11,19,23,35,103,133,293,329,365,511}
d=513,c:2{135,513}
d=515,c:10{7,25,43,103,115,515,1957,2369,19261,35741}
d=517,c:22{47,55,125,173,251,275,379,427,443,515,517,523,571,611,643,715,787,803,871,935,979,1111}
d=519,c:3{21,111,519}
d=521,c:2{55,521}
d=523,c:4{89,115,209,523}
d=525,c:18{87,105,147,159,195,219,255,267,303,309,357,375,429,525,1995,2415,19635,36435}
d=527,c:7{13,31,35,71,221,527,713}
d=529,c:7{29,49,115,161,241,529,943}
d=531,c:8{9,531,1197,1341,1629,1665,1953,1989}
d=533,c:12{13,41,59,533,5371,8651,9307,10291,10619,11603,11767,13079}
d=535,c:10{5,7,47,107,193,535,2033,2461,20009,37129}
d=537,c:2{15,537}
d=539,c:8{7,19,31,49,275,385,539,637}
d=541,c:2{25,541}
d=543,c:4{33,69,165,543}
d=545,c:9{31,49,95,109,545,2071,2507,20383,37823}
d=547,c:4{13,283,305,547}
d=549,c:3{9,549,2115}
d=551,c:16{19,37,59,101,145,209,551,72409,120665,132977,134045,141305,146489,151345,199345,212665}
d=553,c:9{1,7,13,49,239,395,553,1631,1855}
d=555,c:10{3,111,285,345,435,555,2109,2553,20757,38517}
d=557,c:5{5,17,23,47,557}
d=559,c:12{1,13,43,559,5633,9073,9761,10793,11137,12169,12341,13717}
d=561,c:7{15,33,51,129,561,663,759}
d=563,c:2{19,563}
d=565,c:9{5,13,19,113,565,2147,2599,21131,39211}
d=567,c:2{405,567}
d=569,c:2{11,569}
d=571,c:3{19,61,571}
d=573,c:4{141,573,2883,7215}
d=575,c:15{19,31,65,85,115,125,161,175,391,575,1025,2185,2645,21505,39905}
d=577,c:11{17,41,47,61,79,91,107,151,155,217,577}
d=579,c:5{3,15,93,273,579}
d=581,c:6{53,415,455,581,763,1099}
d=583,c:5{7,53,583,689,1133}
d=585,c:16{45,117,171,585,2223,2691,5895,9495,10215,11295,11655,12735,12915,14355,21879,40599}
d=587,c:4{173,181,299,587}
d=589,c:5{29,91,155,247,589}
d=591,c:2{15,591}
d=593,c:5{1,23,29,77,593}
d=595,c:20{5,13,25,29,35,49,55,115,119,137,221,289,425,595,625,805,2261,2737,22253,41293}
d=597,c:3{39,141,597}
d=599,c:4{95,175,205,599}
d=601,c:13{5,7,31,65,103,115,119,131,161,211,313,601,1303}
d=603,c:2{153,603}
d=605,c:15{11,13,25,43,55,77,95,121,451,605,715,2299,2783,22627,41987}
d=607,c:6{13,23,25,65,85,607}
d=609,c:8{3,21,129,231,435,609,80031,148155}
d=611,c:22{47,65,187,325,331,349,515,611,629,845,949,1105,1157,1313,6157,9917,10669,11797,12173,13301,13489,14993}
d=613,c:4{5,11,613,7139}
d=615,c:8{15,123,129,615,2337,2829,23001,42681}
d=617,c:4{19,383,547,617}
d=619,c:2{67,619}
d=621,c:4{135,189,621,1107}
d=623,c:5{43,119,145,445,623}
d=625,c:14{5,29,125,175,235,425,569,625,715,2375,2875,4495,23375,43375}
d=627,c:8{3,57,165,525,627,705,741,1077}
d=629,c:9{1,37,59,323,391,493,629,787,851}
d=631,c:2{353,631}
d=633,c:2{111,633}
d=635,c:10{5,19,127,205,635,893,2413,2921,23749,44069}
d=637,c:16{1,7,49,175,325,413,455,637,6419,10339,11123,12299,12691,13867,14063,15631}
d=639,c:8{261,279,639,23265,25281,35865,37089,39681}
d=641,c:2{7,641}
d=643,c:7{1,7,19,37,77,379,643}
d=645,c:10{3,15,93,129,141,645,2451,2967,24123,44763}
d=647,c:4{7,65,493,647}
d=649,c:13{11,25,59,97,649,767,1063,1463,1639,1991,2035,2387,2431}
d=651,c:5{51,177,273,465,651}
d=653,c:2{5,653}
d=655,c:10{7,65,85,115,131,655,2489,3013,24497,45457}
d=657,c:4{45,171,423,657}
d=659,c:3{71,293,659}
d=661,c:3{17,455,661}
d=663,c:13{15,39,51,663,897,6681,10761,11577,12801,13209,14433,14637,16269}
d=665,c:17{7,23,55,119,133,139,161,175,247,295,323,475,665,2527,3059,24871,46151}
d=667,c:18{23,145,181,203,211,221,253,275,281,293,329,443,569,635,667,1189,87653,162265}
d=669,c:3{105,213,669}
d=671,c:11{11,61,89,151,161,167,169,295,671,793,2585}
d=673,c:4{1,13,31,673}
d=675,c:8{135,189,459,675,2565,3105,25245,46845}
d=677,c:3{119,301,677}
d=679,c:8{7,23,59,91,103,485,679,1009}
d=681,c:2{15,681}
d=683,c:2{35,683}
d=685,c:13{5,13,31,137,205,685,2515,2603,3151,3715,4835,25619,47539}
d=687,c:9{21,57,69,87,93,111,141,159,687}
d=689,c:12{53,79,689,1339,6943,11183,12031,13303,13727,14999,15211,16907}
d=691,c:2{31,691}
d=693,c:5{9,63,495,693,819}
d=695,c:9{13,31,55,139,695,2641,3197,25993,48233}
d=697,c:15{1,7,17,19,25,41,47,67,79,227,263,647,665,697,943}
d=699,c:20{15,699,1941,2181,2757,2949,3045,3237,3333,3477,3525,3669,4101,4317,4605,4665,4953,4965,4989,5721}
d=701,c:2{83,701}
d=703,c:7{71,97,185,361,437,551,703}
d=705,c:16{33,51,57,75,141,375,705,975,1095,1275,1335,1515,2679,3243,26367,48927}
d=707,c:11{25,47,49,77,133,161,203,217,259,505,707}
d=709,c:2{203,709}
d=711,c:5{9,63,711,2097,2385}
d=713,c:7{1,155,217,299,401,713,1271}
d=715,c:28{7,13,35,53,55,59,65,85,91,143,145,209,389,533,715,845,2717,3289,7205,11605,12485,13805,14245,15565,15785,17545,26741,49621}
d=717,c:5{3,39,57,339,717}
d=719,c:7{1,215,433,583,625,673,719}
d=721,c:7{11,35,73,145,161,515,721}
d=723,c:3{21,177,723}
d=725,c:19{5,11,25,31,49,115,145,157,203,235,275,493,725,2755,3335,27115,50315,95275,176375}
d=727,c:6{49,79,89,125,727,1577}
d=729,c:1{729}
d=731,c:6{17,35,43,275,731,989}
d=733,c:4{13,37,437,733}
d=735,c:12{129,147,273,357,375,501,525,735,2793,3381,27489,51009}
d=737,c:5{23,67,187,737,871}
d=739,c:2{1,739}
d=741,c:17{3,15,21,33,57,195,435,741,1011,7467,12027,12939,14307,14763,16131,16359,18183}
d=743,c:4{13,31,313,743}
d=745,c:21{19,31,47,49,71,95,97,101,107,121,149,169,311,329,745,2831,3335,3427,3463,27863,51703}
d=747,c:4{585,747,981,1413}
d=749,c:5{1,7,61,535,749}
d=751,c:2{13,751}
d=753,c:3{273,753,4335}
d=755,c:16{101,103,151,205,389,413,565,637,737,755,833,869,2869,3473,28237,52397}
d=757,c:3{43,85,757}
d=759,c:9{3,39,51,69,165,231,759,897,1353}
d=761,c:2{71,761}
d=763,c:5{29,65,133,545,763}
d=765,c:9{45,63,153,765,1035,2907,3519,28611,53091}
d=767,c:19{7,13,35,59,767,1729,1937,2353,2405,2821,2873,7729,12449,13393,14809,15281,16697,16933,18821}
d=769,c:4{23,41,131,769}
d=771,c:4{3,771,1101,1905}
d=773,c:3{31,515,773}
d=775,c:11{5,17,155,217,325,527,775,2945,3565,28985,53785}
d=777,c:12{87,177,201,255,321,339,363,399,483,555,609,777}
d=779,c:6{5,17,19,29,205,779}
d=781,c:41{29,71,83,107,119,157,203,211,223,227,239,251,259,283,287,319,323,331,341,347,367,373,383,395,421,431,437,439,485,493,503,557,581,653,781,923,28435,30899,43835,45331,48499}
d=783,c:5{27,297,783,102897,190485}
d=785,c:13{11,17,43,65,97,113,157,785,2983,2987,3611,29359,54479}
d=787,c:3{17,317,787}
d=789,c:4{21,51,111,789}
d=791,c:5{7,31,59,565,791}
d=793,c:14{13,61,161,259,793,3055,7991,12871,13847,15311,15799,17263,17507,19459}
d=795,c:10{141,147,159,201,795,1545,3021,3657,29733,55173}
d=797,c:3{5,157,797}
d=799,c:12{1,47,85,425,503,799,1081,1105,1241,1445,1513,1717}
d=801,c:2{153,801}
d=803,c:15{13,17,25,35,37,41,43,55,65,73,83,209,517,803,949}
d=805,c:20{1,37,55,91,95,119,125,161,175,245,299,391,395,575,805,1435,3059,3703,30107,55867}
d=807,c:19{165,417,537,633,669,681,753,777,807,849,861,957,969,993,1041,1101,1149,1185,1293}
d=809,c:2{83,809}
d=811,c:6{1,35,67,77,79,811}
d=813,c:2{51,813}
d=815,c:12{31,47,49,55,85,103,163,815,3097,3749,30481,56561}
d=817,c:7{7,19,79,187,215,295,817}
d=819,c:14{9,63,225,531,585,819,8253,13293,14301,15813,16317,17829,18081,20097}
d=821,c:2{1,821}
d=823,c:3{1,43,823}
d=825,c:14{3,15,75,105,165,231,561,615,825,975,3135,3795,30855,57255}
d=827,c:8{161,193,217,229,341,419,637,827}
d=829,c:3{1,91,829}
d=831,c:2{39,831}
d=833,c:14{7,23,31,35,49,77,121,161,425,595,833,875,1127,1711}
d=835,c:15{1,23,65,157,167,187,193,277,475,635,835,3173,3841,31229,57949}
d=837,c:2{351,837}
d=839,c:5{31,35,71,91,839}
d=841,c:6{29,31,319,841,110519,204595}
d=843,c:8{3,51,57,75,111,123,843,861}
d=845,c:19{1,55,65,85,169,247,845,3211,3887,8515,13715,14755,16315,16835,18395,18655,20735,31603,58643}
d=847,c:8{5,11,35,77,133,605,847,1001}
d=849,c:4{15,159,849,2463}
d=851,c:11{19,49,185,259,437,529,667,851,1517,4109,4193}
d=853,c:2{49,853}
d=855,c:10{9,153,171,207,225,855,3249,3933,31977,59337}
d=857,c:2{25,857}
d=859,c:3{163,181,859}
d=861,c:9{15,21,93,363,507,579,615,687,861}
d=863,c:4{7,55,127,863}
d=865,c:15{1,13,35,41,43,79,97,173,185,865,959,3287,3979,32351,60031}
d=867,c:5{3,39,51,867,1173}
d=869,c:9{5,11,29,77,79,869,1027,2563,2915}
d=871,c:12{5,67,221,871,8777,14137,15209,16817,17353,18961,19229,21373}
d=873,c:3{9,117,873}
d=875,c:23{7,13,145,175,245,265,325,329,365,425,445,505,515,595,625,715,875,1001,3325,4025,6293,32725,60725}
d=877,c:2{7,877}
d=879,c:2{33,879}
d=881,c:7{5,17,19,49,73,83,881}
d=883,c:3{19,35,883}
d=885,c:30{15,177,843,885,1803,1995,2091,2211,2235,2283,2427,2451,2643,2667,2715,2775,2811,2931,2967,3027,3099,3255,3315,3351,3363,3639,3687,4071,33099,61419}
d=887,c:5{25,31,361,385,887}
d=889,c:12{5,7,31,287,635,889,967,1007,1159,1571,2575,2663}
d=891,c:3{81,891,1053}
d=893,c:32{35,79,95,235,367,475,619,667,691,707,715,739,803,815,893,947,1075,1135,1163,1171,1199,1235,1243,1291,1343,1379,1387,1451,1471,1615,1691,1919}
d=895,c:9{7,25,31,179,895,3401,4117,33473,62113}
d=897,c:16{15,57,69,195,273,591,897,1599,9039,14559,15663,17319,17871,19527,19803,22011}
d=899,c:8{19,31,59,341,377,899,118141,218705}
d=901,c:7{53,299,379,479,901,1219,1751}
d=903,c:5{21,57,201,645,903}
d=905,c:11{19,55,115,139,181,275,905,3439,4163,33847,62807}
d=907,c:4{35,55,175,907}
d=909,c:8{63,99,171,207,261,279,333,909}
d=911,c:5{53,83,115,137,911}
d=913,c:8{23,65,83,715,913,1079,1199,1727}
d=915,c:11{15,21,33,39,183,915,3477,3525,4209,34221,63501}
d=917,c:6{23,91,119,161,655,917}
d=919,c:5{79,91,97,133,919}
d=921,c:8{15,51,123,411,615,879,921,1059}
d=923,c:19{7,29,71,377,403,923,9301,14981,16117,17821,18389,20093,20377,22649,33605,36517,51805,53573,57317}
d=925,c:20{5,43,73,103,185,197,223,259,287,359,413,475,575,629,725,925,3515,4255,34595,64195}
d=927,c:3{45,207,927}
d=929,c:4{11,35,119,929}
d=931,c:8{23,77,245,413,475,665,931,9517}
d=933,c:3{39,87,933}
d=935,c:18{17,25,41,55,73,77,85,119,187,215,697,935,1105,1265,3553,4301,34969,64889}
d=937,c:8{5,19,35,49,59,65,145,937}
d=939,c:5{15,33,105,141,939}
d=941,c:2{53,941}
d=943,c:29{7,23,29,37,65,73,79,85,89,101,103,119,121,125,133,143,151,157,175,179,197,205,215,221,223,239,287,943,1681}
d=945,c:9{189,351,459,675,945,3591,4347,35343,65583}
d=947,c:2{31,947}
d=949,c:15{1,17,65,73,247,611,949,9563,15403,16571,18323,18907,20659,20951,23287}
d=951,c:3{3,951,1617}
d=953,c:2{7,953}
d=955,c:11{11,13,191,235,955,3629,4393,4805,12025,35717,66277}
d=957,c:8{33,87,111,363,957,1131,125763,232815}
d=959,c:13{5,7,13,19,29,53,83,287,685,959,3521,5201,6769}
d=961,c:4{1,29,403,961}
d=963,c:2{9,963}
d=965,c:19{5,25,37,43,77,109,131,155,163,193,221,343,455,613,965,3667,4439,36091,66971}
d=967,c:5{1,13,47,55,967}
d=969,c:5{15,57,255,969,1311}
d=971,c:2{25,971}
d=973,c:6{19,77,89,131,695,973}
d=975,c:20{75,195,249,273,285,381,663,975,3705,4485,9825,15825,17025,18825,19425,21225,21525,23925,36465,67665}
d=977,c:2{5,977}
d=979,c:5{7,89,187,979,1157}
d=981,c:2{171,981}
d=983,c:4{77,269,343,983}
d=985,c:11{1,17,25,31,47,197,985,3743,4531,36839,68359}
d=987,c:13{3,33,105,525,705,885,987,1221,1365,1533,1785,1869,2121}
d=989,c:8{7,23,37,73,215,301,989,1763}
d=991,c:7{1,991,2689,3001,4025,4961,6481}
d=993,c:3{39,93,993}
d=995,c:9{11,65,199,235,995,3781,4577,37213,69053}
d=997,c:14{19,23,29,31,37,47,49,53,65,79,85,97,133,997}
d=999,c:4{513,621,783,999}