Problems & Puzzles: Puzzles

Puzzle 1225 Palindromic Echo-Primes Giorgos Kalogeropoulos sent the following puzzle: Echo-Primes (A383907) are primes p where the largest prime factor (LPF) of p−1 is a trailing suffix of p. i.e. Prime p=833293 is Echo-Prime because p-1=833292=2*2*3*3*79*293 and the LPF of p-1 is q=293 and q is at the end of p=833293. In other words the LPF of the previous number echoes at the end of p. Here are the first Echo-Primes: 13, 73, 127, 163, 193, 197, 313, 337, 419, 433, 757, 929, 1153, 2017, 2311, 2593, 2647, 3137, 3659, 4483, 4673, 5741, 6857, 7057, 12071, 12097, 13267, 13313, 13619, 14407, 15877... Here are some examples of Palindromic Echo-Primes (PEP) p (PEP) -------> q (LPF of p-1) 3814183 83 182616281 281 361171163 163 10172427101 101 (p & q are both palindromic) 10915351901 1901 Q1. What is the largest PEP that you can find? Q2. Find other examples where both p and q are palindromic primes.

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From July 13 to 19, 2025, contributions came from Emmanuel Vantieghem, Michael Branicky, Gennady Gusev, Paul Cleary, Simon Cavegn, Oscar Volpatti

***

Emmanuel wrote:

I found 101 echo palprimes below  10^17 .
I tabled them in the form {p ,q} :
(3-digit p) ;{313,13},{757,7},{929,29},
(7-digit p) :{3814183,83},
(9-digit p) :{182616281,281},{361171163,163},
(11-digit p) :{10172427101,101},{10915351901,1901},{33134943133,43133},{73346164337,4337},
(13-digit p) :
{1345679765431,5431},{1523112113251,251},{1654169614561,4561},{1702654562071,71},
{1831317131381,131381},{1918586858191,8191},{3086307036803,6803},{3283865683823,3823},
{3599641469953,953},{7772265622777,2777},{9509491949059,9059},
(15-digit p) :{104021696120401,120401},{108468151864801,4801},{157352949253751,751},
{166179373971661,661},{175757868757571,571},{181811414118181,18181},
{188697737796881,881},{376549050945673,45673},{710214070412017,2017},
{729568797865927,5927},{734932323239437,9437},{737807666708737,8737},
{738897202798837,8837},{756888232888657,88657},{928988282889829,829},
{977846888648779,8779},{998812696218899,18899},{10043211111234001,4001},
17-digit p :
{10045609390654001,4001},{10096543834569001,9001},{10182927172928101,8101},
{10642323832324601,601},{10767702120776701,6701},{10777804540877701,701},
{11201328582310211,2310211},{12332289598223321,23321},{12492343234329421,9421},
{13437679897673431,431},{14273714341737241,241},{14397299999279341,9341},
{14903449594430941,30941},{15221913131912251,12251},{15234270707243251,3251},
{15304341114340351,40351},{16351814941815361,15361},{16756067476065761,65761},
{17230705850703271,3271},{17381769496718371,18371},{17803208980230871,30871},
{19636657975663691,63691},{31314915151941313,313},{31618388188381613,1613},
{32473485458437423,37423},{33166264046266133,6133},{33472126362127433,7433},
{33986144044168933,4168933},{35005746564750053,50053},{35381601710618353,18353},
{36484845154848463,48463},{37424361716342473,42473},{37640449694404673,4673},
{37642148984124673,4673},{37794669896649773,773},{38938452425483983,83983},
{70113419191431107,1431107},{70946872227864907,907},{71427392029372417,2417},
{71645157775154617,54617},{73582353535328537,28537},{73769443634496737,6737},
{73918805950881937,81937},{73982756465728937,937},{75585488688458557,557},
{76732197779123767,3767},{76857442924475867,5867},{77459767676795477,5477},
{77578729692787577,7577},{77911502920511977,977},{78826346964362887,2887},
{90687258185278609,8609},{90873581918537809,809},{92115219891251129,1129},
{93059006060095039,5039},{93598272127289539,9539},{93953634243635939,5939},
{94759823032895749,5749},{95364812921846359,6359},{96610823332801669,1669},
{98646700800764689,764689},{98726656565662789,2789}

There are only two new cases in which  q  is also a palprime :
   181811414118181 - 1 = (2^2)*3*5*(7^2)*11*373*829*18481
   31314915151941313 - 1 = (2^6)*(3^2)*17*37*41*127*181*293*313

Later he added:

I worked a bit further on Puzzle 1225  and found three more solutions for  Q2 :

   7404781546451874047 - 1 = 2*7*11*19*1181*3533*8191*74047

   7577039018109307757 - 1 = (2^2)*23*37*41*53*107*109*157*739*757
   9294212775772124929 - 1 = (2^8)*(3^2)*7*(13^2)*73*131*463*829*929

 

***

Michael wrote:
I found the following palindromic echo-primes below 7*10^18.
Four examples where both p and q are palindromic primes are marked.

p (PEP) -------> q (LPF of p-1)
313 13
757 7 (p & q are both palindromic)
929 29
3814183 83
182616281 281
361171163 163
10172427101 101 (p & q are both palindromic)
10915351901 1901
33134943133 43133
73346164337 4337
1345679765431 5431
1523112113251 251
1654169614561 4561
1702654562071 71
1831317131381 131381
1918586858191 8191
3086307036803 6803
3283865683823 3823
3599641469953 953
7772265622777 2777
9509491949059 9059
104021696120401 120401
108468151864801 4801
157352949253751 751
166179373971661 661
175757868757571 571
181811414118181 18181 (p & q are both palindromic)
188697737796881 881
376549050945673 45673
710214070412017 2017
729568797865927 5927
734932323239437 9437
737807666708737 8737
738897202798837 8837
756888232888657 88657
928988282889829 829
977846888648779 8779
998812696218899 18899
10043211111234001 4001
10045609390654001 4001
10096543834569001 9001
10182927172928101 8101
10642323832324601 601
10767702120776701 6701
10777804540877701 701
11201328582310211 2310211
12332289598223321 23321
12492343234329421 9421
13437679897673431 431
14273714341737241 241
14397299999279341 9341
14903449594430941 30941
15221913131912251 12251
15234270707243251 3251
15304341114340351 40351
16351814941815361 15361
16756067476065761 65761
17230705850703271 3271
17381769496718371 18371
17803208980230871 30871
19636657975663691 63691
31314915151941313 313 (p & q are both palindromic)
31618388188381613 1613
32473485458437423 37423
33166264046266133 6133
33472126362127433 7433
33986144044168933 4168933
35005746564750053 50053
35381601710618353 18353
36484845154848463 48463
37424361716342473 42473
37640449694404673 4673
37642148984124673 4673
37794669896649773 773
38938452425483983 83983
70113419191431107 1431107
70946872227864907 907
71427392029372417 2417
71645157775154617 54617
73582353535328537 28537
73769443634496737 6737
73918805950881937 81937
73982756465728937 937
75585488688458557 557
76732197779123767 3767
76857442924475867 5867
77459767676795477 5477
77578729692787577 7577
77911502920511977 977
78826346964362887 2887
90687258185278609 8609
90873581918537809 809
92115219891251129 1129
93059006060095039 5039
93598272127289539 9539
93953634243635939 5939
94759823032895749 5749
95364812921846359 6359
96610823332801669 1669
98646700800764689 764689
98726656565662789 2789
1003144226224413001 3001
1003190101010913001 3001
1004672132312764001 4001
1014586406046854101 54101
1045134825284315401 15401
1063976778776793601 93601
1069078632368709601 601
1069425675765249601 9601
1075931252521395701 5701
1080491645461940801 40801
1082611324231162801 62801
1084672520252764801 4801
1085247714177425801 5801
1115151814181515111 515111
1139266205026629311 629311
1213113970793113121 13121
1213332825282333121 3121
1224438144418344221 44221
1227518723278157221 57221
1229618506058169221 69221
1255225894985225521 5521
1267792073702977621 77621
1271388774778831721 31721
1273475198915743721 43721
1274944300034494721 4721
1285722163612275821 5821
1324153938393514231 4231
1349089780879809431 9431
1352415740475142531 2531
1359613430343169531 169531
1394203502053024931 4931
1399147911197419931 7419931
1403366216126633041 3041
1430519727279150341 50341
1432567049407652341 2341
1443532235322353441 53441
1460232091902320641 641
1460784506054870641 641
1487494400044947841 7841
1493753810183573941 941
1499199157519919941 9941
1502822030302282051 82051
1521446909096441251 251
1526317358537136251 36251
1564669257529664651 4651
1564686825286864651 4651
1566105037305016651 16651
1570121600061210751 751
1573141461641413751 13751
1587257866687527851 27851
1587736941496377851 377851
1597889625269887951 7951
1605222618162225061 225061
1612499960699942161 2161
1618032752572308161 8161
1633604281824063361 3361
1654741103011474561 4561
1659227628267229561 229561
1661308047408031661 661
1669588529258859661 9661
1684620468640264861 4861
1715608512158065171 65171
1726307555557036271 6271
1733706829286073371 3371
1741754704074571471 71471
1763664984894663671 3671
1795173744473715971 971
1801435344435341081 41081
1822697900097962281 2281
1827949125219497281 281
1841787533357871481 1481
1841947940497491481 1481
1841965567655691481 1481
1844503862683054481 4481
1846114546454116481 6481
1846396711176936481 6481
1847716117116177481 177481
1861646363636461681 61681
1867315218125137681 7681
1882809135319082881 881
1895684123214865981 5981
1906421462641246091 46091
1913650516150563191 3191
1935130665660315391 15391
1939761627261679391 9391
3004882850582884003 884003
3065286372736825603 25603
3068337376737338603 38603
3086996459546996803 6803
3088126317136218803 8803
3114215172715124113 24113
3115919628269195113 5113
3145408875788045413 5413
3221453459543541223 1223
3284188220228814823 823
3286715012105176823 6823
3305243996993425033 25033
3322979691969792233 92233
3364472448442744633 44633
3398019345439108933 8933
3424291010101924243 4243
3438772409042778343 2778343
3452284458544822543 2543
3458735277725378543 8543
3472192002002912743 12743
3483897252527983843 83843
3484420114110244843 244843
3517638193918367153 67153
3533371311131733353 33353
3538195357535918353 18353
3594285913195824953 24953
3638502646462058363 8363
3676001729271006763 6763
3731558647468551373 1373
3742465074705642473 2473
3775115787875115773 773
3801196155516911083 11083
3907740891980477093 77093
3924499946499944293 44293
3953526968696253593 3593
3976435887885346793 5346793

Gennady wrote:

Q1. 94759823032895749, 5749

Q2. 31314915151941313, 313

***

Here are a few more where both are palindromes..

757 --LPF is-- 7  --both are palindromes--
10172427101 --LPF is-- 101  --both are palindromes--
181811414118181 --LPF is-- 18181  --both are palindromes--
31314915151941313 --LPF is-- 313  --both are palindromes--
7404781546451874047 --LPF is-- 74047  --both are palindromes--
7577039018109307757 --LPF is-- 757  --both are palindromes--
9294212775772124929 --LPF is-- 929  --both are palindromes--

and here is my list of with just a PEP the last one being the largest found in the time aloud..

313 --LPF is-- 13
757 --LPF is-- 7
929 --LPF is-- 29
3814183 --LPF is-- 83
182616281 --LPF is-- 281
361171163 --LPF is-- 163
10172427101 --LPF is-- 101
10915351901 --LPF is-- 1901
33134943133 --LPF is-- 43133
73346164337 --LPF is-- 4337
1345679765431 --LPF is-- 5431
1523112113251 --LPF is-- 251
1654169614561 --LPF is-- 4561
1702654562071 --LPF is-- 71
1831317131381 --LPF is-- 131381
1918586858191 --LPF is-- 8191
3086307036803 --LPF is-- 6803
3283865683823 --LPF is-- 3823
3599641469953 --LPF is-- 953
7772265622777 --LPF is-- 2777
9509491949059 --LPF is-- 9059
104021696120401 --LPF is-- 120401
108468151864801 --LPF is-- 4801
157352949253751 --LPF is-- 751
166179373971661 --LPF is-- 661
175757868757571 --LPF is-- 571
181811414118181 --LPF is-- 18181
188697737796881 --LPF is-- 881
376549050945673 --LPF is-- 45673
710214070412017 --LPF is-- 2017
729568797865927 --LPF is-- 5927
734932323239437 --LPF is-- 9437
737807666708737 --LPF is-- 8737
738897202798837 --LPF is-- 8837
756888232888657 --LPF is-- 88657
928988282889829 --LPF is-- 829
977846888648779 --LPF is-- 8779
998812696218899 --LPF is-- 18899
10043211111234001 --LPF is-- 4001
10045609390654001 --LPF is-- 4001
10096543834569001 --LPF is-- 9001
10182927172928101 --LPF is-- 8101
10642323832324601 --LPF is-- 601
10767702120776701 --LPF is-- 6701
10777804540877701 --LPF is-- 701
11201328582310211 --LPF is-- 2310211
12332289598223321 --LPF is-- 23321
12492343234329421 --LPF is-- 9421
13437679897673431 --LPF is-- 431
14273714341737241 --LPF is-- 241
14397299999279341 --LPF is-- 9341
14903449594430941 --LPF is-- 30941
15221913131912251 --LPF is-- 12251
15234270707243251 --LPF is-- 3251
15304341114340351 --LPF is-- 40351
16351814941815361 --LPF is-- 15361
16756067476065761 --LPF is-- 65761
17230705850703271 --LPF is-- 3271
17381769496718371 --LPF is-- 18371
17803208980230871 --LPF is-- 30871
19636657975663691 --LPF is-- 63691
31314915151941313 --LPF is-- 313
31618388188381613 --LPF is-- 1613
32473485458437423 --LPF is-- 37423
33166264046266133 --LPF is-- 6133
33472126362127433 --LPF is-- 7433
33986144044168933 --LPF is-- 4168933
35005746564750053 --LPF is-- 50053
35381601710618353 --LPF is-- 18353
36484845154848463 --LPF is-- 48463
37424361716342473 --LPF is-- 42473
37640449694404673 --LPF is-- 4673
37642148984124673 --LPF is-- 4673
37794669896649773 --LPF is-- 773
38938452425483983 --LPF is-- 83983
70113419191431107 --LPF is-- 1431107
70946872227864907 --LPF is-- 907
71427392029372417 --LPF is-- 2417
71645157775154617 --LPF is-- 54617
73582353535328537 --LPF is-- 28537
73769443634496737 --LPF is-- 6737
73918805950881937 --LPF is-- 81937
73982756465728937 --LPF is-- 937
75585488688458557 --LPF is-- 557
76732197779123767 --LPF is-- 3767
76857442924475867 --LPF is-- 5867
77459767676795477 --LPF is-- 5477
77578729692787577 --LPF is-- 7577
77911502920511977 --LPF is-- 977
78826346964362887 --LPF is-- 2887
90687258185278609 --LPF is-- 8609
90873581918537809 --LPF is-- 809
92115219891251129 --LPF is-- 1129
93059006060095039 --LPF is-- 5039
93598272127289539 --LPF is-- 9539
93953634243635939 --LPF is-- 5939
94759823032895749 --LPF is-- 5749
95364812921846359 --LPF is-- 6359
96610823332801669 --LPF is-- 1669
98646700800764689 --LPF is-- 764689
98726656565662789 --LPF is-- 2789
1003144226224413001 --LPF is-- 3001
1003190101010913001 --LPF is-- 3001
1004672132312764001 --LPF is-- 4001
1014586406046854101 --LPF is-- 54101
1045134825284315401 --LPF is-- 15401
1063976778776793601 --LPF is-- 93601
1069078632368709601 --LPF is-- 601
1069425675765249601 --LPF is-- 9601
1075931252521395701 --LPF is-- 5701
1080491645461940801 --LPF is-- 40801
1082611324231162801 --LPF is-- 62801
1084672520252764801 --LPF is-- 4801
1085247714177425801 --LPF is-- 5801
1115151814181515111 --LPF is-- 515111
1139266205026629311 --LPF is-- 629311
1213113970793113121 --LPF is-- 13121
1213332825282333121 --LPF is-- 3121
1224438144418344221 --LPF is-- 44221
1227518723278157221 --LPF is-- 57221
1229618506058169221 --LPF is-- 69221
1255225894985225521 --LPF is-- 5521
1267792073702977621 --LPF is-- 77621
1271388774778831721 --LPF is-- 31721
1273475198915743721 --LPF is-- 43721
1274944300034494721 --LPF is-- 4721
1285722163612275821 --LPF is-- 5821
1324153938393514231 --LPF is-- 4231
1349089780879809431 --LPF is-- 9431
1352415740475142531 --LPF is-- 2531
1359613430343169531 --LPF is-- 169531
1394203502053024931 --LPF is-- 4931
1399147911197419931 --LPF is-- 7419931
1403366216126633041 --LPF is-- 3041
1430519727279150341 --LPF is-- 50341
1432567049407652341 --LPF is-- 2341
1443532235322353441 --LPF is-- 53441
1460232091902320641 --LPF is-- 641
1460784506054870641 --LPF is-- 641
1487494400044947841 --LPF is-- 7841
1493753810183573941 --LPF is-- 941
1499199157519919941 --LPF is-- 9941
1502822030302282051 --LPF is-- 82051
1521446909096441251 --LPF is-- 251
1526317358537136251 --LPF is-- 36251
1564669257529664651 --LPF is-- 4651
1564686825286864651 --LPF is-- 4651
1566105037305016651 --LPF is-- 16651
1570121600061210751 --LPF is-- 751
1573141461641413751 --LPF is-- 13751
1587257866687527851 --LPF is-- 27851
1587736941496377851 --LPF is-- 377851
1597889625269887951 --LPF is-- 7951
1605222618162225061 --LPF is-- 225061
1612499960699942161 --LPF is-- 2161
1618032752572308161 --LPF is-- 8161
1633604281824063361 --LPF is-- 3361
1654741103011474561 --LPF is-- 4561
1659227628267229561 --LPF is-- 229561
1661308047408031661 --LPF is-- 661
1669588529258859661 --LPF is-- 9661
1684620468640264861 --LPF is-- 4861
1715608512158065171 --LPF is-- 65171
1726307555557036271 --LPF is-- 6271
1733706829286073371 --LPF is-- 3371
1741754704074571471 --LPF is-- 71471
1763664984894663671 --LPF is-- 3671
1795173744473715971 --LPF is-- 971
1801435344435341081 --LPF is-- 41081
1822697900097962281 --LPF is-- 2281
1827949125219497281 --LPF is-- 281
1841787533357871481 --LPF is-- 1481
1841947940497491481 --LPF is-- 1481
1841965567655691481 --LPF is-- 1481
1844503862683054481 --LPF is-- 4481
1846114546454116481 --LPF is-- 6481
1846396711176936481 --LPF is-- 6481
1847716117116177481 --LPF is-- 177481
1861646363636461681 --LPF is-- 61681
1867315218125137681 --LPF is-- 7681
1882809135319082881 --LPF is-- 881
1895684123214865981 --LPF is-- 5981
1906421462641246091 --LPF is-- 46091
1913650516150563191 --LPF is-- 3191
1935130665660315391 --LPF is-- 15391
1939761627261679391 --LPF is-- 9391
3004882850582884003 --LPF is-- 884003
3065286372736825603 --LPF is-- 25603
3068337376737338603 --LPF is-- 38603
3086996459546996803 --LPF is-- 6803
3088126317136218803 --LPF is-- 8803
3114215172715124113 --LPF is-- 24113
3115919628269195113 --LPF is-- 5113
3145408875788045413 --LPF is-- 5413
3221453459543541223 --LPF is-- 1223
3284188220228814823 --LPF is-- 823
3286715012105176823 --LPF is-- 6823
3305243996993425033 --LPF is-- 25033
3322979691969792233 --LPF is-- 92233
3364472448442744633 --LPF is-- 44633
3398019345439108933 --LPF is-- 8933
3424291010101924243 --LPF is-- 4243
3438772409042778343 --LPF is-- 2778343
3452284458544822543 --LPF is-- 2543
3458735277725378543 --LPF is-- 8543
3472192002002912743 --LPF is-- 12743
3483897252527983843 --LPF is-- 83843
3484420114110244843 --LPF is-- 244843
3517638193918367153 --LPF is-- 67153
3533371311131733353 --LPF is-- 33353
3538195357535918353 --LPF is-- 18353
3594285913195824953 --LPF is-- 24953
3638502646462058363 --LPF is-- 8363
3676001729271006763 --LPF is-- 6763
3731558647468551373 --LPF is-- 1373
3742465074705642473 --LPF is-- 2473
3775115787875115773 --LPF is-- 773
3801196155516911083 --LPF is-- 11083
3907740891980477093 --LPF is-- 77093
3924499946499944293 --LPF is-- 44293
3953526968696253593 --LPF is-- 3593
3976435887885346793 --LPF is-- 5346793
7125509120219055217 --LPF is-- 55217
7139687238327869317 --LPF is-- 69317
7146221443441226417 --LPF is-- 26417
7165777970797775617 --LPF is-- 617
7264610317130164627 --LPF is-- 164627
7265637888887365627 --LPF is-- 365627
7320423026203240237 --LPF is-- 40237
7328857676767588237 --LPF is-- 8237
7388068381838608837 --LPF is-- 8837
7401895704075981047 --LPF is-- 81047
7404781546451874047 --LPF is-- 74047
7487572990992757847 --LPF is-- 57847
7507255627265527057 --LPF is-- 7057
7523629587859263257 --LPF is-- 3257
7541128535358211457 --LPF is-- 211457
7543231553551323457 --LPF is-- 3457
7543231864681323457 --LPF is-- 3457
7564139069609314657 --LPF is-- 4657
7577039018109307757 --LPF is-- 757
7586186824286816857 --LPF is-- 6857
7587059377739507857 --LPF is-- 857
7587650052500567857 --LPF is-- 567857
7589492732372949857 --LPF is-- 9857
7672674771774762767 --LPF is-- 2767
7707502490942057077 --LPF is-- 57077
7751248864688421577 --LPF is-- 21577
7785192004002915877 --LPF is-- 15877
7835430727270345387 --LPF is-- 5387
7844441083801444487 --LPF is-- 444487
7850561172711650587 --LPF is-- 587
7903637982897363097 --LPF is-- 63097
7927676279726767297 --LPF is-- 7297
7994851985891584997 --LPF is-- 997
7999738068608379997 --LPF is-- 379997
9008378562658738009 --LPF is-- 8009
9008767941497678009 --LPF is-- 8009
9050262563652620509 --LPF is-- 20509
9073311727271133709 --LPF is-- 133709
9104435252525344019 --LPF is-- 4019
9167771806081777619 --LPF is-- 777619
9214507415147054129 --LPF is-- 4129
9217609485849067129 --LPF is-- 7129
9224891339331984229 --LPF is-- 4229
9236321939391236329 --LPF is-- 236329
9238031142411308329 --LPF is-- 8329
9294212775772124929 --LPF is-- 929
9299108224228019929 --LPF is-- 9929
9329282149412829239 --LPF is-- 9239
9340933051503390439 --LPF is-- 439
9370194515154910739 --LPF is-- 10739
9379771722271779739 --LPF is-- 739
9426217265627126249 --LPF is-- 26249
9464622804082264649 --LPF is-- 4649
9468726379736278649 --LPF is-- 78649
9478132095902318749 --LPF is-- 18749
9479453941493549749 --LPF is-- 9749
9484375133315734849 --LPF is-- 34849
9551897543457981559 --LPF is-- 81559
9609824149414289069 --LPF is-- 89069
9622762005002672269 --LPF is-- 2269
9691178622268711969 --LPF is-- 11969
9721121787871211279 --LPF is-- 11279
9753190161610913579 --LPF is-- 913579
9795451076701545979 --LPF is-- 45979
9802343052503432089 --LPF is-- 2089
9808012310132108089 --LPF is-- 8089
9821687588857861289 --LPF is-- 1289
9956710557550176599 --LPF is-- 176599
10016381397179318361001 --LPF is-- 361001
10018520948084902581001 --LPF is-- 81001
10020110922122901102001 --LPF is-- 102001
10036914653635641963001 --LPF is-- 3001
10042363603030636324001 --LPF is-- 24001
10042696168886169624001 --LPF is-- 24001
10045488162426188454001 --LPF is-- 54001
10049675003930057694001 --LPF is-- 4001
10061293658485639216001 --LPF is-- 16001
10061886239993268816001 --LPF is-- 16001
10069759459895495796001 --LPF is-- 96001
10070463394549336407001 --LPF is-- 7001
10071124979497942117001 --LPF is-- 7001
10074763378087336747001 --LPF is-- 7001
10075971503630517957001 --LPF is-- 7001
10081301788688710318001 --LPF is-- 10318001
10082913423732431928001 --LPF is-- 28001
10091449945654994419001 --LPF is-- 9001
10091979566466597919001 --LPF is-- 19001
10092266581918566229001 --LPF is-- 9001
10096692172027129669001 --LPF is-- 9001
10096878560706587869001 --LPF is-- 69001
10100667827472876600101 --LPF is-- 600101
10144934482928443944101 --LPF is-- 44101
10145653701110735654101 --LPF is-- 54101
10147448833233884474101 --LPF is-- 74101
10149911671017611994101 --LPF is-- 1994101
10154343187878134345101 --LPF is-- 5101
10157654793439745675101 --LPF is-- 5101
10167936403030463976101 --LPF is-- 6101
10168837874947873886101 --LPF is-- 6101
10192445492629454429101 --LPF is-- 429101

***

Simon wrote:

Q1 Largest 3 I found
1919990267300000000037620999191, 99191
303786776800000050000008677687303, 3
79717214685800000600000858641271797, 1271797

Q2
10172427101, 101
181811414118181, 18181
31314915151941313, 313
9294212775772124929, 929
7404781546451874047, 74047
7577039018109307757, 757
143416735989537614341, 14341
160611085303580116061, 16061
184814799898997418481, 18481
30403264317871346230403, 30403
10501614536963541610501, 10501
3010388859465649588830103, 30103
303786776800000050000008677687303, 3

***

Oscar wrote:

I attempted an exaustive search up to 10^21, but my program crashed at about 9.2*10^20.

I found about 800 solutions, I'm sending most interesting pairs (p,q). 

largest p:

914923641292146329419  329419

largest q:

355948558767855849553  55849553

largest ratio log(p)/log(q):

142584608909806485241  241

smallest ratio log(p)/log(q):

929  29

palprime q:

757  7

10172427101  101

181811414118181  18181

31314915151941313  313

7404781546451874047  74047

7577039018109307757  757

9294212775772124929  929

143416735989537614341  14341

160611085303580116061  16061

184814799898997418481  18481

 

***