Problems & Puzzles: Puzzles

Puzzle 1164  181601808106181 As a follow up from previous Puzzle 1163, both Oscar Volpatti & Emmanuel Vantieghem noticed the following solution: 181601808106181 -> 181901808109181 = 101 * 1801008001081 Oscar added: "In these cases, both factors of resulting semiprime are tetradic primes." As a matter of fact, Oscar found a second and larger example: 10608808080880601 -> 10908808080880901 = 101 *108008000800801 Q. Find more examples like these two

---

From Feb 24 to March 1, contributions came from Michael Branicky, PaulCleary, Giorgos Kalogeropoulos, JM Rebert, Emmanuel Vantieghem, Oscar Volpatti

***

Michael wrote:

These are easy to find for increasingly large numbers of digits. Here are ten:

 

  21 digits: 111110618606816011111 -> 111110918909819011111 = 101 * 1100108108018010011
  23 digits: 11161868881018886816111 -> 11191898881018889819111 = 101 * 110810880010088018011
  25 digits: 1018081068860688601808101 -> 1018081098890988901808101 = 101 * 10080010880108801008001
  27 digits: 101011668810101018866110101 -> 101011998810101018899110101 = 101 * 1000118800100010088110001
  29 digits: 10100081661800000816618000101 -> 10100081991800000819918000101 = 101 * 100000811800000008118000001
  31 digits: 1010008088866160616688808000101 -> 1010008088899190919988808000101 = 101 * 10000080088110801188008000001
  33 digits: 101000081661880606088166180000101 -> 101000081991880909088199180000101 = 101 * 1000000811800801080081180000001
  35 digits: 10100000011618106060181611000000101 -> 10100000011918109090181911000000101 = 101 * 100000000118001080100811000000001
  37 digits: 1010000000106080080800806010000000101 -> 1010000000109080080800809010000000101 = 101 * 10000000001080000800008010000000001
  39 digits: 101000000010688601000106886010000000101 -> 101000000010988901000109889010000000101 = 101 * 1000000000108801000001088010000000001

 

***

Paul wrote:

Here is an updated list that supersedes all previous submissions on this subject, sorry they are not in any particular order.  The Tetradic numbers containing only the digits {0,1}  are all palindromes up to 41 digits and those containing the digits {0,1,8} are up to 31 digits in length.  There are some 203 solutions ranging from 17 to 71 digit prime palindromes.

 

1  45  111866101668111010081000180010111866101668111 --> 111899101998111010081000180010111899101998111 = 1008100018001 x 111000000000000010000000000000111
2  17  10608808080880601 --> 10908808080880901 = 101 x 108008000800801
3  53  11011008868011116160016061016061006161111086880011011 --> 11011008898011119190019091019091009191111089880011011 = 10010008080001001 x 1100000001000001000001000001000000011
4  21  181888681000186888181 --> 181888981000189888181 = 101 x 1800881000001880081
5  57  111001168808661101161008011000110800161101166808861100111 --> 111001198808991101191008011000110800191101199808891100111 = 1000010800080100001 x 111000000000100000000000001000000000111
6  59  11800116180000008110166611000000011666101180000008161100811 --> 11800119180000008110199911000000011999101180000008191100811 = 1180000000000000811 x 10000101000000000000100000000000010100001
7  59  11111108881886011611180001001010010008111611068818880111111 --> 11111108881889011911180001001010010008111911098818880111111 = 1001000800080001001 x 11100000001000000000000000000010000000111
8  23  11666868880008886866611 --> 11999898880008889899911 = 101 x 118810880000088018811
9  23  11668808080008080886611 --> 11998808080008080889911 = 101 x 118800080000080008811
10  25  1161810180800080810181611 --> 1191810180800080810181911 = 101 x 11800100800000800100811
11  25  1161810608000008060181611 --> 1191810908000008090181911 = 101 x 11800108000000080100811
12  25  1161800081810181800081611 --> 1191800081810181800081911 = 101 x 11800000810001800000811
13  37  1060800106011161600061611106010080601 --> 1090800109011191900091911109010080901 = 101000010000101 x 10800000010001000000801
14  55  1161800081810181618081616100016161808161810181800081611 --> 1191800081810181918081919100019191808191810181800081911 = 11800000810001800000811 x 101000000000000010000000000000101
15  55  1060800001010101108080601100011060808011010101000080601 --> 1090800001010101108080901100011090808011010101000080901 = 10800000010001000000801 x 101000000000000010000000000000101
16  57  101011818186100161616861611101116168616161001681818110101 --> 101011818189100191919891911101119198919191001981818110101 = 10000081000000018000001 x 10101100000000001010000000000110101
17  59  10000010018081018061001000016061000010016081018081001000001 --> 10000010018081018091001000019091000010019081018081001000001 = 10000000018081000000001 x 1000001000000000001000000000001000001
18  61  1101111166011886160611161101110111011611160616881106611111011 --> 1101111199011889190911191101110111011911190919881109911111011 = 10010010800000801001001 x 110001000100000000000000000001000100011
19  63  101110181668000818661010111000000000111010166818000866181011101 --> 101110181998000818991010111000000000111010199818000899181011101 = 10000018000000081000001 x 10111000000000000000000000000000000011101
20  27  111188860100000001068881111 --> 111188890100000001098881111 = 101 x 1100880100000000010880011
21  27  111618608010101010806816111 --> 111918908010101010809819111 = 101 x 1108108000100010008018011
22  27  111608106080000080601806111 --> 111908109080000080901809111 = 101 x 1108001080000000801008011
23  27  111688616160101061616886111 --> 111988919190101091919889111 = 101 x 1108801180100010811088011
24  27  116181060800000008060181611 --> 119181090800000008090181911 = 101 x 1180010800000000080100811
25  27  101186681010101010186681101 --> 101189981010101010189981101 = 101 x 1001881000100010001881001
26  27  101011668810101018866110101 --> 101011998810101018899110101 = 101 x 1000118800100010088110001
27  27  101888681611101116186888101 --> 101888981911101119189888101 = 101 x 1008801801100011081088001
28  39  106181010810106181000181601018010181601 --> 109181010810109181000181901018010181901 = 101000010000101 x 1081000000000000000001801
29  45  110008801110000811166000661118000011108800011 --> 110008801110000811199000991118000011108800011 = 110000000010000000011 x 1000080010000000100800001
30  51  100018010811810611161616606616161116018118010810001 --> 100018010811810911191919909919191119018118010810001 = 1000080000010100000800001 x 100010010010000010010010001
31  57  101000010180808061010001106000601100010160808081010000101 --> 101000010180808091010001109000901100010190808081010000101 = 1000000100800080010000001 x 101000000000000010000000000000101
32  57  101000818688808866818081186000681180818668808886818000101 --> 101000818988808899818081189000981180818998808889818000101 = 1000008108800088018000001 x 101000000000000010000000000000101
33  61  1100000001861166000811181611180811161811180006611681000000011 --> 1100000001891199000811181911180811191811180009911981000000011 = 1000000000810180000000001 x 1100000001000001000001000001000000011
34  63  101018181818111010661618661111606111166816166010111818181810101 --> 101018181818111010991918991111909111199819199010111818181810101 = 1000080010000000100800001 x 101010100000010000010000010000001010101
35  29  11168666666618081666666686111 --> 11198999999918081999999989111 = 101 x 110881188118000811881188011
36  29  11168681816110001161818686111 --> 11198981819110001191818989111 = 101 x 110881008110000011800188011
37  29  11616068861618081616886061611 --> 11919098891918081919889091911 = 101 x 118010880118000811088010811
38  29  11618181808888088880818181611 --> 11918181808888088880818181911 = 101 x 118001800088000880008100811
39  29  11618816118181018181161881611 --> 11918819118181018181191881911 = 101 x 118008110081000180011800811
40  29  10100081661800000816618000101 --> 10100081991800000819918000101 = 101 x 100000811800000008118000001
41  29  10108888186880008868188880101 --> 10108888189880008898188880101 = 101 x 100088001880000088100880001
42  29  10181661616180008161616618101 --> 10181991919180008191919918101 = 101 x 100811801180000081108118001
43  29  10180601116688088661110608101 --> 10180901119988088991110908101 = 101 x 100801001188000881100108001
44  29  10180606886818081868860608101 --> 10180909889818081898890908101 = 101 x 100801088018000810880108001
45  29  10188881116181018161118888101 --> 10188881119181018191118888101 = 101 x 100880011081000180110088001
46  29  10618180800808080800808181601 --> 10918180800808080800808181901 = 101 x 108100800008000800008001801
47  29  10618606010000000001060681601 --> 10918909010000000001090981901 = 101 x 108108010000000000010801801
48  29  10608001818101010181810080601 --> 10908001818101010181810080901 = 101 x 108000018001000100810000801
49  29  10608000000808080800000080601 --> 10908000000808080800000080901 = 101 x 108000000008000800000000801
50  29  10688881860601010606818888601 --> 10988881890901010909818888901 = 101 x 108800810801000108018008801
51  29  18161666810101010101866616181 --> 18191999810101010101899919181 = 101 x 180118810001000100018811081
52  29  18160111668601010686611106181 --> 18190111998901010989911109181 = 101 x 180100118801000108811001081
53  29  18168188186080008068188186181 --> 18198188189080008098188189181 = 101 x 180180081080000080180081081
54  29  18180818688601010688681808181 --> 18180818988901010988981808181 = 101 x 180008108801000108801800081
55  29  18666666616181018161666666681 --> 18999999919181018191999999981 = 101 x 188118811081000180118811881
56  29  18688186666180008166668188681 --> 18988189999180008199998188981 = 101 x 188001881180000081188100881
57  55  1000010100081116800161116860686111610086111800010100001 --> 1000010100081119800191119890989111910089111800010100001 = 100000010008000800010000001 x 10000100000001110000000100001
58  55  1108011160811080011081166160616611801100801180611108011 --> 1108011190811080011081199190919911801100801180911108011 = 110800000000000000000008011 x 10000101000100000100010100001
59  59  11161810008080006188101816611011661810188160008080001816111 --> 11191810008080009188101819911011991810188190008080001819111 = 110810000080000080000018011 x 101000000000000010000000000000101
60  63  100000100100808101608001001100606001100100806101808001001000001 --> 100000100100808101908001001100909001100100809101808001001000001 = 100000000100808001000000001 x 1000001000000000001000000000001000001
61  63  110811080110811081116086116611606116611680611180118011080118011 --> 110811080110811081119089119911909119911980911180118011080118011 = 110800000000000000000008011 x 1000100001000100010100010001000010001
62  67  1108001108000110801108160611188110118811160618011080110008011008011 --> 1108001108000110801108190911188110118811190918011080110008011008011 = 110800000000000000000008011 x 10000010000001000010001000010000001000001
63  67  1111166660001800000008666111161010101611116668000000081000666611111 --> 1111199990001800000008999111191010101911119998000000081000999911111 = 100018000000000000000810001 x 11110000100000000000000000000000100001111
64  31  1111818188860100010688818181111 --> 1111818188890100010988818181111 = 101 x 11008100880100000108800180011
65  31  1111888601018180818101068881111 --> 1111888901018180818101098881111 = 101 x 11008801000180008100010880011
66  31  1116161618111110111118161616111 --> 1119191918111110111118191919111 = 101 x 11081108100110001100180118011
67  31  1116166666160800080616666616111 --> 1119199999190800080919999919111 = 101 x 11081188110800000801188118011
68  31  1116868688880800080888868686111 --> 1119898988880800080888898989111 = 101 x 11088108800800000800880188011
69  31  1116888818616110116168188886111 --> 1119888818919110119198188889111 = 101 x 11088008108110001180180088011
70  31  1116888808018180818108088886111 --> 1119888808018180818108088889111 = 101 x 11088008000180008100080088011
71  31  1161601816618000008166181061611 --> 1191901819918000008199181091911 = 101 x 11801008118000000081180010811
72  31  1166681000181800081810001866611 --> 1199981000181800081810001899911 = 101 x 11881000001800000810000018811
73  31  1166881816161800081616181886611 --> 1199881819191800081919181889911 = 101 x 11880018011800000811081008811
74  31  1166880116668100018666110886611 --> 1199880119998100018999110889911 = 101 x 11880001188100000188110008811
75  31  1166880888818180818188880886611 --> 1199880888818180818188880889911 = 101 x 11880008800180008100880008811
76  31  1011166860100000000010686611101 --> 1011199890100000000010989911101 = 101 x 10011880100000000000108811001
77  31  1011018866616010106166688101101 --> 1011018899919010109199988101101 = 101 x 10010088118010001081188001001
78  31  1010081661881810181881661800101 --> 1010081991881810181881991800101 = 101 x 10000811800810001800811800001
79  31  1010081811061810181601181800101 --> 1010081811091810181901181800101 = 101 x 10000810010810001801001800001
80  31  1018161111111010101111111618101 --> 1018191111111010101111111918101 = 101 x 10081100110010001001100118001
81  31  1018181081868880888681801818101 --> 1018181081898880888981801818101 = 101 x 10081000810880008801800018001
82  31  1018068180010100010100818608101 --> 1018098180010100010100818908101 = 101 x 10080180000100000100008108001
83  31  1061818061160800080611608181601 --> 1091818091190800080911908181901 = 101 x 10810080110800000801108001801
84  31  1060818688018180818108868180601 --> 1090818988018180818108898180901 = 101 x 10800188000180008100088100801
85  31  1060808081068880888601808080601 --> 1090808081098880888901808080901 = 101 x 10800080010880008801008000801
86  31  1060880601808000008081060880601 --> 1090880901808000008081090880901 = 101 x 10800801008000000080010800801
87  31  1068661188880000000888811668601 --> 1098991188880000000888811998901 = 101 x 10881100880000000008800118801
88  31  1068681816111010101116181868601 --> 1098981819111010101119181898901 = 101 x 10881008110010001001180018801
89  31  1068860110101010101010110688601 --> 1098890110101010101010110988901 = 101 x 10880100100010001000100108801
90  31  1068881011161800081611101888601 --> 1098881011191800081911101888901 = 101 x 10880010011800000811001008801
91  31  1816011068666180816668601106181 --> 1819011098999180819998901109181 = 101 x 18010010881180008118801001081
92  31  1816018060616110116160608106181 --> 1819018090919110119190908109181 = 101 x 18010080108110001180108001081
93  31  1816060801061810181601080606181 --> 1819090801091810181901080909181 = 101 x 18010800010810001801000801081
94  31  1816816010088880888800106186181 --> 1819819010088880888800109189181 = 101 x 18018010000880008800001081081
95  31  1816816068661100011668606186181 --> 1819819098991100011998909189181 = 101 x 18018010881100000118801081081
96  31  1818080801068880888601080808181 --> 1818080801098880888901080808181 = 101 x 18000800010880008801000800081
97  31  1818818686661800081666868188181 --> 1818818989991800081999898188181 = 101 x 18008108811800000811880180081
98  31  1818808116188080808816118088181 --> 1818808119188080808819118088181 = 101 x 18008001180080008008110080081
99  31  1818886818068180818608186888181 --> 1818889818098180818908189888181 = 101 x 18008810080180008108001880081
100  31  1866818060188880888810608186681 --> 1899818090188880888810908189981 = 101 x 18810080100880008800108001881
101  31  1868606186088080808806816068681 --> 1898909189088080808809819098981 = 101 x 18801081080080008008018010881
102  31  1868818668608000008068668188681 --> 1898818998908000008098998188981 = 101 x 18800188108000000080188100881
103  43  1161601118061661686800086861661608111061611 --> 1191901118091991989800089891991908111091911 = 101000010000101 x 11801000000800000800000010811
104  47  18688001868688001868800000886810088686810088681 --> 18988001898988001898800000889810088989810088981 = 1010000101010000101 x 18800000000000000000000000881
105  47  18688018818818801868800000886810881881881088681 --> 18988018818818801898800000889810881881881088981 = 1010001001001000101 x 18800000000000000000000000881
106  51  100001010081100801600618101816006108001180010100001 --> 100001010081100801900918101819009108001180010100001 = 10000100000100000100001 x 10000001008000000080010000001
107  61  1880000001881868801868818888100018888188681088681881000000881 --> 1880000001881898801898818888100018888188981088981881000000881 = 18800000000000000000000000881 x 100000000100101000101001000000001
108  61  1001180180100018100018181066110116601818100018100010810811001 --> 1001180180100018100018181099110119901818100018100010810811001 = 10001800000000000000000810001 x 100100000010000010000010000001001
109  67  1000000001081101080180181161168010108611611810810801011801000000001 --> 1000000001081101080180181191198010108911911810810801011801000000001 = 10000000000801010800000000001 x 100000000100100000010000001001000000001
110  33  111100816668681808186866618001111 --> 111100819998981808189899918001111 = 101 x 1100008118801800081088118000011
111  33  111108068666160808061666860801111 --> 111108098999190808091999890801111 = 101 x 1100080188110800080118810800011
112  33  111181616868100000001868616181111 --> 111181919898100000001898919181111 = 101 x 1100811088100000000018801180011
113  33  111181818061160808061160818181111 --> 111181818091190808091190818181111 = 101 x 1100810080110800080110800180011
114  33  111188661608000000000806166881111 --> 111188991908000000000809199881111 = 101 x 1100881108000000000008011880011
115  33  111188861668810101018866168881111 --> 111188891998810101018899198881111 = 101 x 1100880118800100010088110880011
116  33  111616111868880808088868111616111 --> 111919111898880808088898111919111 = 101 x 1108110018800800080088100118011
117  33  111618608806161808161608806816111 --> 111918908809191808191908809819111 = 101 x 1108108008011800081108008018011
118  33  111618686661800000008166686816111 --> 111918989991800000008199989819111 = 101 x 1108108811800000000081188018011
119  33  111618688801010000010108886816111 --> 111918988801010000010108889819111 = 101 x 1108108800010000000100088018011
120  33  111606060810608000806018060606111 --> 111909090810908000809018090909111 = 101 x 1108010800108000008010080108011
121  33  111608106088888000888880601806111 --> 111908109088888000888880901809111 = 101 x 1108001080088000008800801008011
122  33  111608186880080808080088681806111 --> 111908189880080808080088981809111 = 101 x 1108001880000800080000881008011
123  33  111608886661601000106166688806111 --> 111908889991901000109199988809111 = 101 x 1108008811801000001081188008011
124  33  111688810111618000816111018886111 --> 111988810111918000819111018889111 = 101 x 1108800100118000008110010088011
125  33  111688808060110101011060808886111 --> 111988808090110101011090808889111 = 101 x 1108800080100100010010800088011
126  33  116168180081611000116180081861611 --> 119198180081911000119180081891911 = 101 x 1180180000811000001180000810811
127  33  116181060800010101010008060181611 --> 119181090800010101010008090181911 = 101 x 1180010800000100010000080100811
128  33  116180010101060808060101010081611 --> 119180010101090808090101010081911 = 101 x 1180000100010800080100010000811
129  33  116180018686010000010686810081611 --> 119180018989010000010989810081911 = 101 x 1180000188010000000108810000811
130  33  116180081611808000808116180081611 --> 119180081911808000808119180081911 = 101 x 1180000811008000008001180000811
131  33  116188888101808000808101888881611 --> 119188888101808000808101888881911 = 101 x 1180088001008000008001008800811
132  33  116686161800101000101008161686611 --> 119989191800101000101008191989911 = 101 x 1188011800001000001000081108811
133  33  116688008866160808061668800886611 --> 119988008899190808091998800889911 = 101 x 1188000088110800080118800008811
134  33  116688088681818101818186880886611 --> 119988088981818101818189880889911 = 101 x 1188000881008100018001880008811
135  33  116688088880818101818088880886611 --> 119988088880818101818088880889911 = 101 x 1188000880008100018000880008811
136  33  116688818688810101018886818886611 --> 119988818988810101018889818889911 = 101 x 1188008108800100010088018008811
137  33  101116160686661808166686061611101 --> 101119190989991808199989091911101 = 101 x 1001180108811800081188010811001
138  33  101101816110888808888011618101101 --> 101101819110888808888011918101101 = 101 x 1001008110008800088000118001001
139  33  101181810188681000186881018181101 --> 101181810188981000189881018181101 = 101 x 1001800100881000001880010081001
140  33  101186681080880808088080186681101 --> 101189981080880808088080189981101 = 101 x 1001881000800800080080001881001
141  33  101011668601808000808106866110101 --> 101011998901808000808109899110101 = 101 x 1000118801008000008001088110001
142  33  101018161168880000088861161810101 --> 101018191198880000088891191810101 = 101 x 1000180110880000000880110810001
143  33  101001011868800000008868110100101 --> 101001011898800000008898110100101 = 101 x 1000010018800000000088100100001
144  33  101081611010180808081010116180101 --> 101081911010180808081010119180101 = 101 x 1000811000100800080010001180001
145  33  101081810010618101816010018180101 --> 101081810010918101819010018180101 = 101 x 1000810000108100018010000180001
146  33  101080606061160808061160606080101 --> 101080909091190808091190909080101 = 101 x 1000801080110800080110801080001
147  33  101088681181880808088181186880101 --> 101088981181880808088181189880101 = 101 x 1000881001800800080081001880001
148  33  101818608008080000080800806818101 --> 101818908008080000080800809818101 = 101 x 1008108000080000000800008018001
149  33  101818608808818101818808806818101 --> 101818908808818101818808809818101 = 101 x 1008108008008100018008008018001
150  33  101806818181601000106181818608101 --> 101809818181901000109181818908101 = 101 x 1008018001801000001081008108001
151  33  101806818010618101816010818608101 --> 101809818010918101819010818908101 = 101 x 1008018000108100018010008108001
152  33  101808101011668808866110101808101 --> 101808101011998808899110101808101 = 101 x 1008001000118800088110001008001
153  33  101808816618808000808816618808101 --> 101808819918808000808819918808101 = 101 x 1008008118008000008008118008001
154  33  101888801161880808088161108888101 --> 101888801191880808088191108888101 = 101 x 1008800011800800080081100088001
155  33  106181000008181000181800000181601 --> 109181000008181000181800000181901 = 101 x 1081000000081000001800000001801
156  33  106181081616080000080616180181601 --> 109181081919080000080919180181901 = 101 x 1081000811080000000801180001801
157  33  106061668601888808888106866160601 --> 109091998901888808888109899190901 = 101 x 1080118801008800088001088110801
158  33  106060108061110000011160801060601 --> 109090108091110000011190801090901 = 101 x 1080100080110000000110800010801
159  33  106068688806010000010608886860601 --> 109098988809010000010908889890901 = 101 x 1080188008010000000108008810801
160  33  106081818808808000808808818180601 --> 109081818808808000808808818180901 = 101 x 1080018008008000008008008100801
161  33  106080181888860101068888181080601 --> 109080181888890101098888181080901 = 101 x 1080001800880100010880081000801
162  33  106080018668110101011866810080601 --> 109080018998110101011899810080901 = 101 x 1080000188100100010018810000801
163  33  106080818108080000080801818080601 --> 109080818108080000080801818080901 = 101 x 1080008100080000000800018000801
164  33  106088161110100000001011161880601 --> 109088191110100000001011191880901 = 101 x 1080081100100000000010011800801
165  33  106088181808111101111808181880601 --> 109088181808111101111808181880901 = 101 x 1080081008001100011008001800801
166  33  106088061160611101116061160880601 --> 109088091190911101119091190880901 = 101 x 1080080110801100011080110800801
167  33  106088081060681808186060180880601 --> 109088081090981808189090180880901 = 101 x 1080080010801800081080100800801
168  33  106866118060110101011060811668601 --> 109899118090110101011090811998901 = 101 x 1088110080100100010010800118801
169  33  106866168661100000001166861668601 --> 109899198991100000001199891998901 = 101 x 1088110881100000000011880118801
170  33  106866618888800000008888816668601 --> 109899918888800000008888819998901 = 101 x 1088118008800000000088008118801
171  33  106886060800080808080008060688601 --> 109889090800080808080008090988901 = 101 x 1088010800000800080000080108801
172  33  181611181661880808088166181116181 --> 181911181991880808088199181119181 = 101 x 1801100811800800080081180011081
173  33  181616686868800000008868686616181 --> 181919989898800000008898989919181 = 101 x 1801188018800000000088108811081
174  33  181601010188681000186881010106181 --> 181901010188981000189881010109181 = 101 x 1801000100881000001880010001081
175  33  181606080111608000806111080606181 --> 181909080111908000809111080909181 = 101 x 1801080001108000008011000801081
176  33  181681611181810000018181116186181 --> 181981911181810000018181119189181 = 101 x 1801801100810000000180011081081
177  33  181811110010608000806010011118181 --> 181811110010908000809010011118181 = 101 x 1800110000108000008010000110081
178  33  181811618886860808068688816118181 --> 181811918889890808098988819118181 = 101 x 1800118008810800080188008110081
179  33  181810180880681808186088081018181 --> 181810180880981808189088081018181 = 101 x 1800100800801800081080080010081
180  33  181801116886111000111688611108181 --> 181801119889111000111988911108181 = 101 x 1800011088011000001108801100081
181  33  181801816010181808181010618108181 --> 181801819010181808181010918108181 = 101 x 1800018010001800081000108100081
182  33  181800106888808000808888601008181 --> 181800109888808000808888901008181 = 101 x 1800001088008000008008801000081
183  33  181800186880080808080088681008181 --> 181800189880080808080088981008181 = 101 x 1800001880000800080000881000081
184  33  181800010688611101116886010008181 --> 181800010988911101119889010008181 = 101 x 1800000108801100011088010000081
185  33  181888661101161808161101166888181 --> 181888991101191808191101199888181 = 101 x 1800881100011800081100011880081
186  33  186661688808181000181808886166681 --> 189991988808181000181808889199981 = 101 x 1881108800081000001800088011881
187  33  186681101808808000808808101186681 --> 189981101808808000808808101189981 = 101 x 1881001008008000008008001001881
188  33  186681000806868808868608000186681 --> 189981000809898808898908000189981 = 101 x 1881000008018800088108000001881
189  33  186681008888101000101888800186681 --> 189981008888101000101888800189981 = 101 x 1881000088001000001008800001881
190  33  186681818100080808080001818186681 --> 189981818100080808080001818189981 = 101 x 1881008100000800080000018001881
191  33  186686061110181808181011160686681 --> 189989091110181808181011190989981 = 101 x 1881080110001800081000110801881
192  33  186860108868100000001868801068681 --> 189890108898100000001898801098981 = 101 x 1880100088100000000018800010881
193  33  186868160686661808166686061868681 --> 189898190989991808199989091898981 = 101 x 1880180108811800081188010810881
194  33  186881010001010000010100010188681 --> 189881010001010000010100010188981 = 101 x 1880010000010000000100000100881
195  33  186888061666661808166666160888681 --> 189888091999991808199999190888981 = 101 x 1880080118811800081188110800881
196  51  181800000116000000161601000106161000000611000008181 --> 181800000119000000191901000109191000000911000008181 = 101000000010000000101 x 1800000001000000000001000000081
197  59  10800108080001668800088688806060888688000886610008080100801 --> 10800108080001998800088988809090888988000889910008080100801 = 10000100000001110000000100001 x 1080000008000000000008000000801
198  65  11800810000000011616668166008110001180066186661611000000001800811 --> 11800810000000011919998199008110001180099189991911000000001800811 = 1180081000000000000000001800811 x 10000000000000010101000000000000001
199  65  18000000008001681818668880808868086880808886681818610080000000081 --> 18000000008001981818998880808898089880808889981818910080000000081 = 1800000000800000000080000000081 x 10000000000001101010110000000000001
200  67  1000000001800010101061118181816180816181818111601010100081000000001 --> 1000000001800010101091118181819180819181818111901010100081000000001 = 1000000001800000000081000000001 x 1000000000000010101010100000000000001
201  71  10080010081110881000111000181116681018661118100011100018801118001008001 --> 10080010081110881000111000181119981018991118100011100018801118001008001 = 1008000000100000000010000008001 x 10000010000110000000000000001100001000001
202  71  10810800060880000188080188808066860006866080888108088100008806000801801 --> 10810800090880000188080188808099890009899080888108088100008809000801801 = 1080000008000000000008000000801 x 10010000010000000100000100000001000001001
203  71  11006610610081000000000001680161116061116108610000000000018001601660011 --> 11009910910081000000000001980191119091119108910000000000018001901990011 = 1000810000000000000000000180001 x 11001000100000000000000000000000100010011

***

Giorgos wrote:

If the smallest factor of the semiprime is 101, we can easily find many examples.

I am sending you a txt file with 1111 such examples

 

Here are also some other examples with bigger factors of the semiprime:

 

1016086011100810180011106806101 -> 1019089011100810180011109809101 = 1008001 * 1011000000100010000001101
111868161666010666161868111 -> 111898191999010999191898111 = 111010111 * 1008000001000008001
10018188161016168186161016188181001 -> 10018188191019198189191019188181001 = 10008180001 * 1001000001001001000001001
10000060101060806010106000001 -> 10000090101090809010109000001 = 1000008000001 * 10000010101000001
11088086681888118181188818668088011 -> 11088089981888118181188818998088011 = 100800818008001 * 110000000010000000011
1060800106011161600061611106010080601 -> 1090800109011191900091911109010080901 = 101000010000101 * 10800000010001000000801
106181010810106181000181601018010181601 -> 109181010810109181000181901018010181901 = 101000010000101 * 1081000000000000000001801
1010808180888188181608061818818880818080101 -> 1010808180888188181908091818818880818080101 = 10008000800080001 * 101000010000010000010000101
10011010081066081018066018001011001 -> 10011010081099081018099018001011001 = 10011010001011001 * 1000000008000000001
1011110111011618688088808868161101110111101 -> 1011110111011918988088808898191101110111101 = 10111101110111101 * 100000000000080000000000001
168000001116680861111168086611100000861 -> 198000001119980891111198089911100000891 = 18000000100000081 * 11000000001110000000011
100001010611018161668181866161810116010100001 -> 100001010911018191998181899191810119010100001 = 1000000008000000001 * 100001010111010111010100001
100100100811800610186101681016008118001001001 -> 100100100811800910189101981019008118001001001 = 1000000008000000001 * 100100100011000110001001001
10000600018108108160106180180181000600001 -> 10000900018108108190109180180181000900001 = 1000080001000800001 * 10000100000100000100001
100006100168081610661606166016180861001600001 -> 100009100198081910991909199019180891001900001 = 1000080001000800001 * 100001100010000010001100001
100101111111160180688888886081061111111101001 -> 100101111111190180988888889081091111111101001 = 1001011111111101001 * 100000000000080000000000001
101101001110060660600888006066060011100101101 -> 101101001110090990900888009099090011100101101 = 1011010011100101101 * 100000000000080000000000001
101110101110181668180888081866181011101011101 -> 101110101110181998180888081899181011101011101 = 1011101011101011101 * 100000000000080000000000001
110008801110000811166000661118000011108800011 -> 110008801110000811199000991118000011108800011 = 110000000010000000011 * 1000080010000000100800001
11001001111116801801688888610810861111110010011 -> 11001001111119801801988888910810891111110010011 = 110010011111110010011 * 100000000000080000000000001
11100001111116880011688888611008861111110000111 -> 11100001111119880011988888911008891111110000111 = 111000011111110000111 * 100000000000080000000000001
10001108001680161806661016660816108610080110001 -> 10001108001980191809991019990819108910080110001 = 10000008000100080000001 * 1000110000100010000110001
100018010811810611161616606616161116018118010810001 -> 100018010811810911191919909919191119018118010810001 = 1000080000010100000800001 * 100010010010000010010010001
100010100001060008181000606000181800060100001010001 -> 100010100001090008181000909000181800090100001010001 = 1000101000010100001010001 * 100000000000080000000000001
100101001110181160810688181886018061181011100101001 -> 100101001110181190810988181889018091181011100101001 = 1001010011101011100101001 * 100000000000080000000000001
100111011001060066816600606006618660060100110111001 -> 100111011001090099819900909009918990090100110111001 = 1001110110010100110111001 * 100000000000080000000000001
101100000110181680000168181861000086181011000001101 -> 101100000110181980000198181891000089181011000001101 = 1011000001101011000001101 * 100000000000080000000000001
101100010110181681008168181861800186181011010001101 -> 101100010110181981008198181891800189181011010001101 = 1011000101101011010001101 * 100000000000080000000000001
110001000110186100810086181680018001681011000100011 -> 110001000110189100810089181980018001981011000100011 = 1100010001101011000100011 * 100000000000080000000000001
110100100110186180180186181681081081681011001001011 -> 110100100110189180180189181981081081981011001001011 = 1101001001101011001001011 * 100000000000080000000000001
111010100111166608181066686660181806661111001010111 -> 111010100111199908181099989990181809991111001010111 = 1110101001111111001010111 * 100000000000080000000000001
111100100100188680180161181161081086881001001001111 -> 111100100100188980180191181191081089881001001001111 = 1111001001001001001001111 * 100000000000080000000000001

 

Here is one more example with bigger semiprime-factors
1001011111111111111100011010161161686866888666666111661616668666161661116666668886686861611610101100011111111111111101001 -> 1001011111111111111100011010191191989899888999999111991919998999191991119999998889989891911910101100011111111111111101001 = 10000000000000000000000000000800000000000000000000000000001 * 100101111111111111110001101011111110101100011111111111111101001

 

***

Rebert wrote:

I found 254 examples joined in this attached file bpp1164_RJM.txt.

Least solution with d-digit primes.

 

n° d              x            rot(x)     prime factorization

 

1  15 181601808106181 181901808109181 101 * 1801008001081

2  17 10608808080880601 10908808080880901 101 * 108008000800801

3  21 111110618606816011111 111110918909819011111 101 * 1100108108018010011

4  23 11119896016061069891111 11116869019091096861111 101 * 110068010090010860011

5  25 1011018868190918688101101 1011018898160618988101101 101 * 10010088100600188001001

6  27 101011668810101018866110101 101011998810101018899110101 101 * 1000118800100010088110001

7  29 10000060101060806010106000001 10000090101090809010109000001 1000008000001 * 10000010101000001

8  31 1010001116686090906866111000101 1010001119989060609899111000101 101 * 10000011089000600098011000001

 

 

***

Emmanuel wrote:

I wrote the first  336  solutions in a text file that you can find here.
Here is a small sample of that table :
(I show : { palprime, turn, factor1, factor2}
{181601808106181, 181901808109181, 101, 1801008001081},
{10608808080880601, 10908808080880901, 101, 108008000800801},
{111110618606816011111, 111110918909819011111, 101, 1100108108018010011},
{116666668606866666611, 119999998909899999911, 101, 1188118801088118811},
   .....
{11668860600008080000606886611, 11998890900008080000909889911, 101, 118800900000080000009008811},
{11668861619191019191616886611, 11998891916161016161919889911, 101, 118800910061000160019008811},
{11668866666866066866666886611, 11998899999899099899999889911, 101, 118800990098010890099008811},
{11668888889896069898888886611, 11998888886869096868888889911, 101, 118800880068010860088008811},
{11668898980909090908989886611, 11998868680606060608686889911, 101, 118800680006000600086008811}}

All but two of the solutions have a "turn" of the form  101*m, where  m  is a "turnable" palprime (whose "turn"  can be of various prime signature)
The two exceptions are :
1) {111868161666010666161868111, 111898191999010999191898111, 111010111, 1008000001000008001}
2) {10000060101060806010106000001, 10000090101090809010109000001, 1000008000001, 10000010101000001}

***

Volpatti wrote:

Dear Carlos,

next solution has 21 digits:

P = 111110618606816011111,

S = 111110918909819011111,

S = 101*1100108108018010011.

 

Again, the resulting semiprime S is divisible by 101.

There are 98 solutions with less than 30 digits; semiprime S is coprime with 101 in two cases:
 

 

P = 111868161666010666161868111,

S = 111898191999010999191898111,

S = 111010111*1008000001000008001;
 

 

P = 10000060101060806010106000001,

S = 10000090101090809010109000001,

S = 10000010101000001*1000008000001.
 

 

A titanic solution with 1035 digits can be obtained as follows.

Build number S by concatenating:

114 copies of composite palindrome "1991",

9 zeros,

35 copies of palprime "181",

9 zeros again,

114 copies of composite palindrome "1991" again.

 

As 1991 = 11*181, number S is palindrome and divisible by 181.

Cofactor F = S/181 is a tetradic prime, only containing digits 0 and 1.

Number P obtained by flipping S upside down is a palprime.

 

***