Puzzle 1281 Find the largest prime B such that any block of two adjacent digits in it...

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Puzzle 1281 Find the largest prime such that any block of two adjacent digits in it...

On August 15, 2026, Bernardo Recamán sent the following puzzle:

1. Find the largest prime B such that any block of two adjacent digits in it, is a two-digit number coprime to all other such blocks in it.
2. Do likewise with blocks of three adjacent digits.
3. Redo 1 & 2 if no internal prime-blocks are permitted.

Examples to 1 & 2:

1) 4783531= prime
47= prime
78= 2x3x13
83= prime
35= 5x7
53=prime
31=prime

2) 47835391 = prime
478= 2x239
783=3x3x3x29
835=5x167
353=prime
539=7x7x11
391=17x23

3) 64951
64 = 2^6
49 = 7^2
95 = 5x19
51 = 3x17

4) 998917
998 = 2x499
989 = 23x43
891 = 3^4x11
917 = 7x131

Q. Send your largest pimes for 1, 2, 3 & 4.
____
BTW this is a puzzle alike to Puzzle 253.

From Aug 29 to Set 4, 2026, contributions came from Rahid Zaman, Michael Branicky, Giorgos Kalogeropoulos, Paul Cleary, Michael Hürter

***
Rahid wrote:

For 1281, here's what I have. I don't know if these are the maximum, though I searched and couldn't find anything larger

- Case 1: 8997941735371319

- Case 2: 88779773974337937335997199113713191933173139 (44 digits)

887 Primo
877 Primo
779 19x41
797 Primo
977 Primo
773 Primo
739 Primo
397 Primo
974 2x487
743 Primo
433 Primo
337 Primo
379 Primo
793 13x61
937 Primo
373 Primo
733 Primo
335 5x67
359 Primo
599 Primo
997 Primo
971 Primo
719 Primo
199 Primo
991 Primo
911 Primo
113 Primo
137 Primo
371 7x53
713 23x31
131 Primo
319 11x29
191 Primo
919 Primo
193 Primo
933 3x311
331 Primo
317 Primo
173 Primo
731 17x43
313 Primo
139 Primo

- Case 3: 94951
- Case 4: 97377998951

***
Michael wrote:

1) 8997941735371319 = prime
89 = prime
99 = 9*11
97 = prime
79 = prime
94 = 2*47
41 = prime
17 = prime
73 = prime
35 = 5*7
53 = prime
37 = prime
71 = prime
13 = prime
31 = prime
19 = prime

2) 9613941997971911379311571799193739773313173 = prime (43 digits)

961 = 31*31
613 = 613
139 = 139
394 = 2*197
941 = 941
419 = 419
199 = 199
997 = 997
979 = 11*89
797 = 797
971 = 971
719 = 719
191 = 191
911 = 911
113 = 113
137 = 137
379 = 379
793 = 13*61
931 = 7*7*19
311 = 311
115 = 5*23
157 = 157
571 = 571
717 = 3*239
179 = 179
799 = 17*47
991 = 991
919 = 919
193 = 193
937 = 937
373 = 373
739 = 739
397 = 397
977 = 977
773 = 773
733 = 733
331 = 331
313 = 313
131 = 131
317 = 317
173 = 173

3) 94951 = prime
94 = 2*47
49 = 7*7
95 = 5*19
51 = 3*17

4) 97377998951 = prime
973 = 7*139
737 = 11*67
377 = 13*29
779 = 19*41
799 = 17*47
998 = 2*499
989 = 23*43
895 = 5*179
951 = 3*317

***
Giorgos wrote:

Q1: 2-Digit Blocks, Primes Allowed

8997941735371319 (16 digits)

Q2: 3-Digit Blocks, Primes Allowed

99977397619919719373313173191139533713797 (41 digits)

999 -> {3, 3, 3, 37}
997 -> {997}
977 -> {977}
773 -> {773}
739 -> {739}
397 -> {397}
976 -> {2, 2, 2, 2, 61}
761 -> {761}
619 -> {619}
199 -> {199}
991 -> {991}
919 -> {919}
197 -> {197}
971 -> {971}
719 -> {719}
193 -> {193}
937 -> {937}
373 -> {373}
733 -> {733}
331 -> {331}
313 -> {313}
131 -> {131}
317 -> {317}
173 -> {173}
731 -> {17, 43}
319 -> {11, 29}
191 -> {191}
911 -> {911}
113 -> {113}
139 -> {139}
395 -> {5, 79}
953 -> {953}
533 -> {13, 41}
337 -> {337}
371 -> {7, 53}
713 -> {23, 31}
137 -> {137}
379 -> {379}
797 -> {797}

Q3: 2-Digit Blocks, NO Primes Allowed

94951

Q4: 3-Digit Blocks, NO Primes Allowed

97377998951

***
Paul wrote:

Here are the results with the best I could find..

8997941735371319
88779773974337937335997199113713191933173139
94951
97377998951

Q2. is 44 digits long, 45 digits would require 43 x 3 overlapping digits and an exhaustive search couldn't find one. So I am concluding that 44 digits is the maximum length and the above Prime was the largest I could find.

***
Michael wrote:

Q1:

8997941735371319 (16 digits)

89 89,1;
99 3,2; 11,1;
97 97,1;
79 79,1;
94 2,1; 47,1;
41 41,1;
17 17,1;
73 73,1;
35 5,1; 7,1;
53 53,1;
37 37,1;
71 71,1;
13 13,1;
31 31,1;
19 19,1;

Q2:
9997991977397619379719113733713139533179 (40 digits)

999 3,3; 37,1;
997 997,1;
979 11,1; 89,1;
799 17,1; 47,1;
991 991,1;
919 919,1;
197 197,1;
977 977,1;
773 773,1;
739 739,1;
397 397,1;
976 2,4; 61,1;
761 761,1;
619 619,1;
193 193,1;
937 937,1;
379 379,1;
797 797,1;
971 971,1;
719 719,1;
191 191,1;
911 911,1;
113 113,1;
137 137,1;
373 373,1;
733 733,1;
337 337,1;
371 7,1; 53,1;
713 23,1; 31,1;
131 131,1;
313 313,1;
139 139,1;
395 5,1; 79,1;
953 953,1;
533 13,1; 41,1;
331 331,1;
317 317,1;
179 179,1;

Q3:
94951 (5 digits)

94 2,1; 47,1;
49 7,2;
95 5,1; 19,1;
51 3,1; 17,1;

Q4:

97377998951 (11 digits)

973 7,1; 139,1;
737 11,1; 67,1;
377 13,1; 29,1;
779 19,1; 41,1;
799 17,1; 47,1;
998 2,1; 499,1;
989 23,1; 43,1;
895 5,1; 179,1;
951 3,1; 317,1;

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