Problems & Puzzles: Puzzles

 Puzzle 1107 Place 11 after its base result JM Bergot sent the following nice puzzle, discovered by Robert Israel: I place 11 after its base result:   Base 2: 11=1011 and 101111 is prime. Base 3: 11=102 and  10211 is prime. Base 4: 11=23 and 2311 is prime. Base 5: 11=21 and 2111 is prime. Base 6: 11=15 and 1511 is prime. Base 7: 11=14 and 1411=17*83. Q. Is there a prime that will work for bases 2 to 7 or more?

During the week from 16-21 October, 2022, contributions came from Michael Branicky, Paul Cleary, Giorgos Kalogeropoulos, Emmanuel Vantieghem, Oscar Volpatti

***

Michael wrote:

A prime that works for bases 2..8: 18287682827

These primes work for bases 2..7: 2503190171, 3140691149, 3163538981, 5220731281, 8697116507, 8943915647, 9683242171, 11810714449, 15711327157, 18164509927, 18509593967, 23089068221, 23355134323, 24394684109, 25371672061.

There are many primes other than 11 that work for bases 2..6, starting with 14854627, 53385677, 61002743, ...

***

Paul wrote:

My search was primes up to 97011687217, the 4 x 10^9 th prime.

There were another 253 primes that went to base 6.  I found 23 that went to base 7, and 4 that went to base 8.  Here are the base 7 and 8 solutions.

Base 7.

Base 2: 15711327157=1110101000011101111101001110110101 and 111010100001110111110100111011010115711327157 is Prime

Base 3: 15711327157=1111112221122010002211 and 111111222112201000221115711327157 is Prime

Base 4: 15711327157=32220131331032311 and 3222013133103231115711327157 is Prime

Base 5: 15711327157=224134044432112 and 22413404443211215711327157 is Prime

Base 6: 15711327157=11115004133421 and 1111500413342115711327157 is Prime

Base 7: 15711327157=1064225035351 and 106422503535115711327157 is Prime

Base 8: 15711327157=165035751665 and 16503575166515711327157 is not Prime with factors-> {{3,3},{13,1},{3271,1},{14374421482157117,1}}

Base 2: 18164509927=10000111010101100000110110011100111 and 1000011101010110000011011001110011118164509927 is Prime

Base 3: 18164509927=1201212220201201012221 and 120121222020120101222118164509927 is Prime

Base 4: 18164509927=100322230012303213 and 10032223001230321318164509927 is Prime

Base 5: 18164509927=244200103304202 and 24420010330420218164509927 is Prime

Base 6: 18164509927=12202240253211 and 1220224025321118164509927 is Prime

Base 7: 18164509927=1212063533614 and 121206353361418164509927 is Prime

Base 8: 18164509927=207254066347 and 20725406634718164509927 is not Prime with factors-> {{233,1},{56592827,1},{1571758255997,1}}

Base 2: 18509593967=10001001111010000011111110101101111 and 1000100111101000001111111010110111118509593967 is Prime

Base 3: 18509593967=1202202222001210112022 and 120220222200121011202218509593967 is Prime

Base 4: 18509593967=101033100133311233 and 10103310013331123318509593967 is Prime

Base 5: 18509593967=300401424001332 and 30040142400133218509593967 is Prime

Base 6: 18509593967=12300404500355 and 1230040450035518509593967 is Prime

Base 7: 18509593967=1223453644544 and 122345364454418509593967 is Prime

Base 8: 18509593967=211720376557 and 21172037655718509593967 is not Prime with factors-> {{3,3},{3428057,1},{228744604539653,1}}

Base 2: 23089068221=10101100000001101110011100010111101 and 1010110000000110111001110001011110123089068221 is Prime

Base 3: 23089068221=2012121010011012012212 and 201212101001101201221223089068221 is Prime

Base 4: 23089068221=111200031303202331 and 11120003130320233123089068221 is Prime

Base 5: 23089068221=334241300140341 and 33424130014034123089068221 is Prime

Base 6: 23089068221=14335034510205 and 1433503451020523089068221 is Prime

Base 7: 23089068221=1445111561462 and 144511156146223089068221 is Prime

Base 8: 23089068221=254015634275 and 25401563427523089068221 is not Prime with factors-> {{1559,1},{16293498029200185419,1}}

Base 2: 23355134323=10101110000000100110001000101110011 and 1010111000000010011000100010111001123355134323 is Prime

Base 3: 23355134323=2020021122210202021221 and 202002112221020202122123355134323 is Prime

Base 4: 23355134323=111300010301011303 and 11130001030101130323355134323 is Prime

Base 5: 23355134323=340312403244243 and 34031240324424323355134323 is Prime

Base 6: 23355134323=14421301325511 and 1442130132551123355134323 is Prime

Base 7: 23355134323=1454522236423 and 145452223642323355134323 is Prime

Base 8: 23355134323=256004610563 and 25600461056323355134323 is not Prime with factors-> {{3,2},{43,1},{1999,1},{33092077119080671,1}}

Base 2: 24394684109=10110101110000010010101011011001101 and 1011010111000001001010101101100110124394684109 is Prime

Base 3: 24394684109=2022222002220100111212 and 202222200222010011121224394684109 is Prime

Base 4: 24394684109=112232002111123031 and 11223200211112303124394684109 is Prime

Base 5: 24394684109=344430014342414 and 34443001434241424394684109 is Prime

Base 6: 24394684109=15112354424205 and 1511235442420524394684109 is Prime

Base 7: 24394684109=1522344252005 and 152234425200524394684109 is Prime

Base 8: 24394684109=265602253315 and 26560225331524394684109 is not Prime with factors-> {{3,2},{13,1},{164848591,1},{1377084702647,1}}

Base 2: 25371672061=10111101000010001001111110111111101 and 1011110100001000100111111011111110125371672061 is Prime

Base 3: 25371672061=2102111012021111000211 and 210211101202111100021125371672061 is Prime

Base 4: 25371672061=113220101033313331 and 11322010103331333125371672061 is Prime

Base 5: 25371672061=403430122001221 and 40343012200122125371672061 is Prime

Base 6: 25371672061=15353334552421 and 1535333455242125371672061 is Prime

Base 7: 25371672061=1555506440251 and 155550644025125371672061 is Prime

Base 8: 25371672061=275021176775 and 27502117677525371672061 is not Prime with factors-> {{3,2},{3089,1},{989249224039616261,1}}

Base 2: 30036024727=11011111110010010010101110110010111 and 1101111111001001001010111011001011130036024727 is Prime

Base 3: 30036024727=2212112021002011000021 and 221211202100201100002130036024727 is Prime

Base 4: 30036024727=123332102111312113 and 12333210211131211330036024727 is Prime

Base 5: 30036024727=443003210242402 and 44300321024240230036024727 is Prime

Base 6: 30036024727=21444240130011 and 2144424013001130036024727 is Prime

Base 7: 30036024727=2112214666132 and 211221466613230036024727 is Prime

Base 8: 30036024727=337622256627 and 33762225662730036024727 is not Prime with factors-> {{167,1},{541,1},{4793,1},{77966834699437,1}}

Base 2: 32422403759=11110001100100001101001111010101111 and 1111000110010000110100111101010111132422403759 is Prime

Base 3: 32422403759=10002200120110202121212 and 1000220012011020212121232422403759 is Prime

Base 4: 32422403759=132030201221322233 and 13203020122132223332422403759 is Prime

Base 5: 32422403759=1012400113410014 and 101240011341001432422403759 is Prime

Base 6: 32422403759=22521124345035 and 2252112434503532422403759 is Prime

Base 7: 32422403759=2225312612324 and 222531261232432422403759 is Prime

Base 8: 32422403759=361441517257 and 36144151725732422403759 is not Prime with factors-> {{3,1},{7,1},{13,1},{59,1},{181,1},{4787,1},{2589890339171,1}}

Base 2: 33312582889=11111000001100101011010110011101001 and 1111100000110010101101011001110100133312582889 is Prime

Base 3: 33312582889=10011222121111112211021 and 1001122212111111221102133312582889 is Prime

Base 4: 33312582889=133001211122303221 and 13300121112230322133312582889 is Prime

Base 5: 33312582889=1021211010123024 and 102121101012302433312582889 is Prime

Base 6: 33312582889=23145324132441 and 2314532413244133312582889 is Prime

Base 7: 33312582889=2256342165625 and 225634216562533312582889 is Prime

Base 8: 33312582889=370145326351 and 37014532635133312582889 is not Prime with factors-> {{59659,1},{48067373,1},{12907612327,1}}

Base 2: 33314197183=11111000001101011100100111010111111 and 1111100000110101110010011101011111133314197183 is Prime

Base 3: 33314197183=10011222201112120012221 and 1001122220111212001222133314197183 is Prime

Base 4: 33314197183=133001223210322333 and 13300122321032233333314197183 is Prime

Base 5: 33314197183=1021211413302213 and 102121141330221333314197183 is Prime

Base 6: 33314197183=23145422510211 and 2314542251021133314197183 is Prime

Base 7: 33314197183=2256362001211 and 225636200121133314197183 is Prime

Base 8: 33314197183=370153447277 and 37015344727733314197183 is not Prime with factors-> {{3,1},{585149,1},{21085993896559289,1}}

Base 2: 38812156963=100100001001011000101000010000100011 and 10010000100101100010100001000010001138812156963 is Prime

Base 3: 38812156963=10201011212220212011221 and 1020101121222021201122138812156963 is Prime

Base 4: 38812156963=210021120220100203 and 21002112022010020338812156963 is Prime

Base 5: 38812156963=1113441403010323 and 111344140301032338812156963 is Prime

Base 6: 38812156963=25455143115511 and 2545514311551138812156963 is Prime

Base 7: 38812156963=2542541614366 and 254254161436638812156963 is Prime

Base 8: 38812156963=441130502043 and 44113050204338812156963 is not Prime with factors-> {{23,1},{79,1},{2473,1},{163601,1},{60007024243,1}}

Base 2: 45201325361=101010000110001101010111110100110001 and 10101000011000110101011111010011000145201325361 is Prime

Base 3: 45201325361=11022200011021020112112 and 1102220001102102011211245201325361 is Prime

Base 4: 45201325361=222012031113310301 and 22201203111331030145201325361 is Prime

Base 5: 45201325361=1220033014402421 and 122003301440242145201325361 is Prime

Base 6: 45201325361=32433141135105 and 3243314113510545201325361 is Prime

Base 7: 45201325361=3160062632453 and 316006263245345201325361 is Prime

Base 8: 45201325361=520615276461 and 52061527646145201325361 is not Prime with factors-> {{11,1},{337,1},{8071633,1},{1739934558931,1}}

Base 2: 45252037811=101010001001001110110100110010110011 and 10101000100100111011010011001011001145252037811 is Prime

Base 3: 45252037811=11022210200200200221102 and 1102221020020020022110245252037811 is Prime

Base 4: 45252037811=222021032310302303 and 22202103231030230345252037811 is Prime

Base 5: 45252037811=1220134010202221 and 122013401020222145252037811 is Prime

Base 6: 45252037811=32442152115015 and 3244215211501545252037811 is Prime

Base 7: 45252037811=3161246655251 and 316124665525145252037811 is Prime

Base 8: 45252037811=521116646263 and 52111664626345252037811 is not Prime with factors-> {{3,2},{13,1},{1289,1},{345538280031199247,1}}

Base 2: 49540367543=101110001000110101100000010010110111 and 10111000100011010110000001001011011149540367543 is Prime

Base 3: 49540367543=11201212112221102102112 and 1120121211222110210211249540367543 is Prime

Base 4: 49540367543=232020311200102313 and 23202031120010231349540367543 is Prime

Base 5: 49540367543=1302424313230133 and 130242431323013349540367543 is Prime

Base 6: 49540367543=34431502001235 and 3443150200123549540367543 is Prime

Base 7: 49540367543=3402441114302 and 340244111430249540367543 is Prime

Base 8: 49540367543=561065402267 and 56106540226749540367543 is not Prime with factors-> {{11,1},{61,1},{6367,1},{7351,1},{1786527461249,1}}

Base 2: 49923262999=101110011111101010001000101000010111 and 10111001111110101000100010100001011149923262999 is Prime

Base 3: 49923262999=11202212020102112020201 and 1120221202010211202020149923262999 is Prime

Base 4: 49923262999=232133222020220113 and 23213322202022011349923262999 is Prime

Base 5: 49923262999=1304220323403444 and 130422032340344449923262999 is Prime

Base 6: 49923262999=34533500441331 and 3453350044133149923262999 is Prime

Base 7: 49923262999=3415100506501 and 341510050650149923262999 is Prime

Base 8: 49923262999=563752105027 and 56375210502749923262999 is not Prime with factors-> {{23,1},{103,1},{179,2},{523,1},{1420088351197,1}}

Base 2: 51796173389=110000001111010010101110001001001101 and 11000000111101001010111000100100110151796173389 is Prime

Base 3: 51796173389=11221200202122020022012 and 1122120020212202002201251796173389 is Prime

Base 4: 51796173389=300033102232021031 and 30003310223202103151796173389 is Prime

Base 5: 51796173389=1322034310022024 and 132203431002202451796173389 is Prime

Base 6: 51796173389=35443403431005 and 3544340343100551796173389 is Prime

Base 7: 51796173389=3512362131602 and 351236213160251796173389 is Prime

Base 8: 51796173389=601722561115 and 60172256111551796173389 is not Prime with factors-> {{3,1},{20057418703850598724463,1}}

Base 2: 60791965513=111000100111011110111100011101001001 and 11100010011101111011110001110100100160791965513 is Prime

Base 3: 60791965513=12210220200210211012021 and 1221022020021021101202160791965513 is Prime

Base 4: 60791965513=320213132330131021 and 32021313233013102160791965513 is Prime

Base 5: 60791965513=1444000220344023 and 144400022034402360791965513 is Prime

Base 6: 60791965513=43532154444441 and 4353215444444160791965513 is Prime

Base 7: 60791965513=4251324116026 and 425132411602660791965513 is Prime

Base 8: 60791965513=704736743511 and 70473674351160791965513 is not Prime with factors-> {{7,1},{10067667764451541709359,1}}

Base 2: 73411483181=1000100010111101010100101011000101101 and 100010001011110101010010101100010110173411483181 is Prime

Base 3: 73411483181=21000111011122202000112 and 2100011101112220200011273411483181 is Prime

Base 4: 73411483181=1010113222211120231 and 101011322221112023173411483181 is Prime

Base 5: 73411483181=2200321314430211 and 220032131443021173411483181 is Prime

Base 6: 73411483181=53420314505405 and 5342031450540573411483181 is Prime

Base 7: 73411483181=5206130211521 and 520613021152173411483181 is Prime

Base 8: 73411483181=1042752453055 and 104275245305573411483181 is not Prime with factors-> {{3,1},{34758415101857803827727,1}}

Base 2: 74519166587=1000101011001101100000011111001111011 and 100010101100110110000001111100111101174519166587 is Prime

Base 3: 74519166587=21010100100221212222212 and 2101010010022121222221274519166587 is Prime

Base 4: 74519166587=1011121230003321323 and 101112123000332132374519166587 is Prime

Base 5: 74519166587=2210103401312322 and 221010340131232274519166587 is Prime

Base 6: 74519166587=54122244205335 and 5412224420533574519166587 is Prime

Base 7: 74519166587=5245440315305 and 524544031530574519166587 is Prime

Base 8: 74519166587=1053154037173 and 105315403717374519166587 is not Prime with factors-> {{3,3},{3900570508050908117281,1}}

Base 2: 77253764141=1000111111100101011101110010000101101 and 100011111110010101110111001000010110177253764141 is Prime

Base 3: 77253764141=21101101221120211121202 and 2110110122112021112120277253764141 is Prime

Base 4: 77253764141=1013330223232100231 and 101333022323210023177253764141 is Prime

Base 5: 77253764141=2231203430423031 and 223120343042303177253764141 is Prime

Base 6: 77253764141=55253452135245 and 5525345213524577253764141 is Prime

Base 7: 77253764141=5403264106031 and 540326410603177253764141 is Prime

Base 8: 77253764141=1077453562055 and 107745356205577253764141 is not Prime with factors-> {{3253,1},{88721,1},{261641,1},{1426862977,1}}

Base 2: 87847093937=1010001110100000110000100011010110001 and 101000111010000011000010001101011000187847093937 is Prime

Base 3: 87847093937=22101202012210202012202 and 2210120201221020201220287847093937 is Prime

Base 4: 87847093937=1101310012010122301 and 110131001201012230187847093937 is Prime

Base 5: 87847093937=2414402324001222 and 241440232400122287847093937 is Prime

Base 6: 87847093937=104204552032545 and 10420455203254587847093937 is Prime

Base 7: 87847093937=6226634656331 and 622663465633187847093937 is Prime

Base 8: 87847093937=1216406043261 and 121640604326187847093937 is not Prime with factors-> {{115783,1},{1050591229508544839,1}}

Base 2: 95578304851=1011001000000111010010011000101010011 and 101100100000011101001001100010101001195578304851 is Prime

Base 3: 95578304851=100010201000102221210201 and 10001020100010222121020195578304851 is Prime

Base 4: 95578304851=1121000322103011103 and 112100032210301110395578304851 is Prime

Base 5: 95578304851=3031221021223401 and 303122102122340195578304851 is Prime

Base 6: 95578304851=111524050440031 and 11152405044003195578304851 is Prime

Base 7: 95578304851=6622343121001 and 662234312100195578304851 is Prime

Base 8: 95578304851=1310072230523 and 131007223052395578304851 is not Prime with factors-> {{3,1},{1931,1},{22614745909269045107,1}}

Base 8.

Base 2: 18287682827=10001000010000001111110010100001011 and 1000100001000000111111001010000101118287682827 is Prime

Base 3: 18287682827=1202012111111112200012 and 120201211111111220001218287682827 is Prime

Base 4: 18287682827=101002001332110023 and 10100200133211002318287682827 is Prime

Base 5: 18287682827=244423121322302 and 24442312132230218287682827 is Prime

Base 6: 18287682827=12222400302135 and 1222240030213518287682827 is Prime

Base 7: 18287682827=1215120511361 and 121512051136118287682827 is Prime

Base 8: 18287682827=210201762413 and 21020176241318287682827 is Prime

Base 9: 18287682827=52174445605 and 5217444560518287682827 is not Prime with factors-> {{3,1},{11,1},{19,1},{8321283190619278601,1}}

Base 2: 27158998591=11001010010110011010110111000111111 and 1100101001011001101011011100011111127158998591 is Prime

Base 3: 27158998591=2121002202110002002201 and 212100220211000200220127158998591 is Prime

Base 4: 27158998591=121102303112320333 and 12110230311232033327158998591 is Prime

Base 5: 27158998591=421110200423331 and 42111020042333127158998591 is Prime

Base 6: 27158998591=20250543332331 and 2025054333233127158998591 is Prime

Base 7: 27158998591=1651011451624 and 165101145162427158998591 is Prime

Base 8: 27158998591=312263267077 and 31226326707727158998591 is Prime

Base 9: 27158998591=77082402081 and 7708240208127158998591 is not Prime with factors-> {{239,1},{307,1},{105055540977296267,1}}

Base 2: 48447867329=101101000111101101111100100111000001 and 10110100011110110111110010011100000148447867329 is Prime

Base 3: 48447867329=11122001102012122212122 and 1112200110201212221212248447867329 is Prime

Base 4: 48447867329=231013231330213001 and 23101323133021300148447867329 is Prime

Base 5: 48447867329=1243210123223304 and 124321012322330448447867329 is Prime

Base 6: 48447867329=34131233534025 and 3413123353402548447867329 is Prime

Base 7: 48447867329=3333404035421 and 333340403542148447867329 is Prime

Base 8: 48447867329=550755744701 and 55075574470148447867329 is Prime

Base 9: 48447867329=148042178778 and 14804217877848447867329 is not Prime with factors-> {{7,1},{37,1},{43,1},{124097,1},{10711639935361,1}}

Base 2: 68191565713=111111100000100010001011011110010001 and 11111110000010001000101101111001000168191565713 is Prime

Base 3: 68191565713=20112000101110110101221 and 2011200010111011010122168191565713 is Prime

Base 4: 68191565713=333200202023132101 and 33320020202313210168191565713 is Prime

Base 5: 68191565713=2104124020100323 and 210412402010032368191565713 is Prime

Base 6: 68191565713=51154325525041 and 5115432552504168191565713 is Prime

Base 7: 68191565713=4632564513325 and 463256451332568191565713 is Prime

Base 8: 68191565713=774042133621 and 77404213362168191565713 is Prime

Base 9: 68191565713=215011413357 and 21501141335768191565713 is not Prime with factors-> {{7,1},{3071591619395455937959,1}}

***

Giorgos wrote:

prime 2503190171 works for bases 2 to 7
Base 2: 2503190171 = 10010101001100111010011010011011 and 100101010011001110100110100110112503190171 is prime
Base 3: 2503190171 = 20110110012020101002 and 201101100120201010022503190171 is prime
Base 4: 2503190171 = 2111030322122123 and 21110303221221232503190171 is prime
Base 5: 2503190171 = 20111304041141 and 201113040411412503190171 is prime
Base 6: 2503190171 = 1052220015215 and 10522200152152503190171 is prime
Base 7: 2503190171 = 116013523346 and 1160135233462503190171 is prime
Base 8: 2503190171 = 22514723233 and 225147232332503190171 = 3*3*7* 11* 158993* 501031* 4078409

prime 18287682827 works for bases 2 to 8
Base 2: 18287682827 = 10001000010000001111110010100001011 and 1000100001000000111111001010000101118287682827 is prime

Base 3: 18287682827 = 1202012111111112200012 and 120201211111111220001218287682827 is prime
Base 4: 18287682827 = 101002001332110023 and 10100200133211002318287682827 is prime
Base 5: 18287682827 = 244423121322302 and 24442312132230218287682827 is prime
Base 6: 18287682827 = 12222400302135 and 1222240030213518287682827 is prime
Base 7: 18287682827 = 1215120511361 and 121512051136118287682827 is prime
Base 8: 18287682827 = 210201762413 and 21020176241318287682827 is prime
Base 9:18287682827 = 52174445605 and 5217444560518287682827= 3*11*19*8321283190619278601

***

Emmanuel wrote:

Let's denote by  F(m,b)  the concatenation of  the base b  of  m  with  m.
Then, the smallest prime  p  such that  F(p,b)  is prime for  b = 2, 3, ..., n  is :
3 when  n = 2. This means : 113  is prime)
3 when  n = 3. This means.: 103  is also prime
11 when  n = 4. What this means is seen in JM Bergot's examples
11 when  n = 5.    "        "       "      "     "    "       "           "
11 when  n = 6    "        "       "      "     "    "       "           "
2503190171 when  n = 7.  This means that the following numbers are primes :
100101010011001110100110100110112503190171
201101100120201010022503190171
21110303221221232503190171
201113040411412503190171
10522200152152503190171
1160135233462503190171
(64131663322503190171 = 311*45307*659863*6897521)
18287682827 when  n = 8,  This means that the following numbers are primes :
1000100001000000111111001010000101118287682827
120201211111111220001218287682827
10100200133211002318287682827
24442312132230218287682827
1222240030213518287682827
121512051136118287682827
21020176241318287682827
(5217444560518287682827 = 3*11*19*8321283190619278601)

For  n = 9  I found no prime below  4*10^11.
There cannot exist a prime when  n = 10  since  F(p,p)  is always divisible by  p.

***

Oscar wrote:

I found the smallest prime which works for bases 2 to b exactly, with b <= 8.

b  pmin
2  7
3  3
4  7393
5  218987
6  11
7  2503190171
8  18287682827

No prime below 1.77*10^11 works for bases 2 to 9.
No prime p can work for base 10: the resulting number would be the concatenation of two copies of p itself, composite.

Later, on April 20, 2023, he added:

about puzzle 1107, prime 3400973168069 works for bases 2 to 9.
It produces the following primes:

b  p
2  1100010111110110011100110000101101110001013400973168069
3  1100010101112000002010012023400973168069
4  3011331213030023130113400973168069
5  4212101431123342343400973168069
6  111222151340322453400973168069
7  5004661666435613400973168069
8  613731630267053400973168069
9  130334500210523400973168069

***

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