Problems & Puzzles: Puzzles

Puzzle 1227 Extension to Puzzle 1220
Jeff Heleen sent the following nice puzzle extension:

Here is an extension to puzzle 1220.
Let a, b and c be distinct primes.
Concatenate them such that abc is a prime d.
Find a, b, c, d such that a, b and c are factors of d-1.
 
Ex.
  a        b        c       d            d-1         factors of d-1
  2        3        11     2311         2310         2 × 3 × 5 × 7 × 11
  2        11        3     2113         2112         2^6 × 3 × 11
  2        241      31     224131       224130       2 × 3 × 5 × 31 × 241
  5        7        41     5741         5740         2^2 × 5 × 7 × 41
 11        2         3     1123         1122         2 × 3 × 11 × 17
131        13       11     1311311      1311310      2 × 5 × 7 × 11 × 13 × 131

Q1. Find more alike.
Q2. Can the same be done with >3 distinct primes concatenated?

From June 28 to July 4, contributions came from J. M. Rebert, Michael Branicky, Gennady Gusev, Giorgos Kalogeropoulos, Adam Stinchcombe, Simon Cavegn, Emmanuel Vantieghem, Oscar Volpatti

***

Rebert wrote:

Q1. Find more alike.
 I found:
                        
 primes    d      d-1      factorisation
[2, 3, 11] 2311 2310 = 2 * 3 * 5 * 7 * 11;

[5, 7, 41] 5741 5740 = 2^2 * 5 * 7 * 41;

[2, 11, 3] 2113 2112 = 2^6 * 3 * 11;

[11, 2, 3] 1123 1122 = 2 * 3 * 11 * 17;

[2, 5, 641] 25641 25640 = 2^3 * 5 * 641;

[2, 137, 3] 21373 21372 = 2^2 * 3 * 13 * 137;

[2, 7, 3253] 273253 273252 = 2^2 * 3 * 7 * 3253;

[2, 113, 11] 211311 211310 = 2 * 5 * 11 * 17 * 113;

[2, 241, 31] 224131 224130 = 2 * 3 * 5 * 31 * 241;

[7, 2, 39779] 7239779 7239778 = 2 * 7 * 13 * 39779;

[2, 6173, 53] 2617353 2617352 = 2^3 * 53 * 6173;

[131, 13, 11] 1311311 1311310 = 2 * 5 * 7 * 11 * 13 * 131;

[131, 13, 11] 1311311 1311310 = 2 * 5 * 7 * 11 * 13 * 131;
                       
[29, 5, 12721] 29512721 29512720 = 2^4 * 5 * 29 * 12721;

[131, 5, 5021] 13155021 13155020 = 2^2 * 5 * 131 * 5021;


Q2. Can the same be done with >3 distinct primes concatenated?

I found:
                   
 primes    d      d-1      factorisation
[2, 3, 11] 2311 2310 = 2 * 3 * 5 * 7 * 11;

[5, 7, 41] 5741 5740 = 2^2 * 5 * 7 * 41;

[2, 11, 3] 2113 2112 = 2^6 * 3 * 11;

[11, 2, 3] 1123 1122 = 2 * 3 * 11 * 17;

[5, 2, 17, 31] 521731 521730 = 2 * 3^2 * 5 * 11 * 17 * 31;

[5, 2, 17, 31] 521731 521730 = 2 * 3^2 * 5 * 11 * 17 * 31;

[2, 19, 3, 37] 219337 219336 = 2^3 * 3 * 13 * 19 * 37;

[2, 73, 3, 13] 273313 273312 = 2^5 * 3^2 * 13 * 73;

[2, 19, 3, 37] 219337 219336 = 2^3 * 3 * 13 * 19 * 37;

[2, 73, 3, 13] 273313 273312 = 2^5 * 3^2 * 13 * 73;

[11, 2, 5, 31] 112531 112530 = 2 * 3 * 5 * 11^2 * 31;

[13, 7, 2, 29] 137229 137228 = 2^2 * 7 * 13^2 * 29;

[19, 3, 11, 7] 193117 193116 = 2^2 * 3 * 7 * 11^2 * 19;

[7, 2, 131, 23] 7213123 7213122 = 2 * 3^2 * 7 * 19 * 23 * 131;

[251, 2, 5, 11] 2512511 2512510 = 2 * 5 * 7 * 11 * 13 * 251;

[271, 2, 7, 11] 2712711 2712710 = 2 * 5 * 7 * 11 * 13 * 271;

[521, 5, 2, 11] 5215211 5215210 = 2 * 5 * 7 * 11 * 13 * 521;

[571, 5, 7, 11] 5715711 5715710 = 2 * 5 * 7 * 11 * 13 * 571;

[751, 7, 5, 11] 7517511 7517510 = 2 * 5 * 7 * 11 * 13 * 751;

[3, 61, 13, 23] 3611323 3611322 = 2 * 3^2 * 11 * 13 * 23 * 61;

[3, 61, 13, 23] 3611323 3611322 = 2 * 3^2 * 11 * 13 * 23 * 61;

[3, 61, 13, 23] 3611323 3611322 = 2 * 3^2 * 11 * 13 * 23 * 61;

[3, 61, 13, 23] 3611323 3611322 = 2 * 3^2 * 11 * 13 * 23 * 61;

[3, 61, 13, 23] 3611323 3611322 = 2 * 3^2 * 11 * 13 * 23 * 61;

[3, 61, 13, 23] 3611323 3611322 = 2 * 3^2 * 11 * 13 * 23 * 61;

[17, 13, 19, 3] 1713193 1713192 = 2^3 * 3 * 13 * 17^2 * 19;

[17, 13, 19, 3] 1713193 1713192 = 2^3 * 3 * 13 * 17^2 * 19;

[17, 13, 19, 3] 1713193 1713192 = 2^3 * 3 * 13 * 17^2 * 19;

[5, 2, 7, 20921] 52720921 52720920 = 2^3 * 3^2 * 5 * 7 * 20921;

[3, 2, 22567, 7] 32225677 32225676 = 2^2 * 3 * 7 * 17 * 22567;

[2, 2713, 23, 7] 22713237 22713236 = 2^2 * 7 * 13 * 23 * 2713;

[2, 3, 661, 157] 23661157 23661156 = 2^2 * 3 * 19 * 157 * 661;

[3, 2, 271, 223] 32271223 32271222 = 2 * 3 * 89 * 223 * 271;

[2, 3, 661, 157] 23661157 23661156 = 2^2 * 3 * 19 * 157 * 661;

[3, 2, 271, 223] 32271223 32271222 = 2 * 3 * 89 * 223 * 271;

[2, 3, 661, 157] 23661157 23661156 = 2^2 * 3 * 19 * 157 * 661;

[3, 2, 271, 223] 32271223 32271222 = 2 * 3 * 89 * 223 * 271;

[2, 3, 661, 157] 23661157 23661156 = 2^2 * 3 * 19 * 157 * 661;

[3, 2, 271, 223] 32271223 32271222 = 2 * 3 * 89 * 223 * 271;

[3, 2, 5, 7, 11] 325711 325710 = 2 * 3^2 * 5 * 7 * 11 * 47;

[3, 2, 5, 7, 11] 325711 325710 = 2 * 3^2 * 5 * 7 * 11 * 47;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[7, 3, 2, 13, 41, 31] 732134131 732134130 = 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

[5, 3, 13, 19, 2, 31] 531319231 531319230 = 2 * 3^3 * 5 * 13 * 19 * 31 * 257;

...

Q1. Find more alike.
 I found:
                        
 primes    d    d-1    factorisation
[2, 3, 11] 2311 2310 = 2 * 3 * 5 * 7 * 11;

[5, 7, 41] 5741 5740 = 2^2 * 5 * 7 * 41;

[2, 11, 3] 2113 2112 = 2^6 * 3 * 11;

[11, 2, 3] 1123 1122 = 2 * 3 * 11 * 17;
[2, 5, 641] 25641 25640 = 2^3 * 5 * 641;

[2, 137, 3] 21373 21372 = 2^2 * 3 * 13 * 137;

[2, 7, 3253] 273253 273252 = 2^2 * 3 * 7 * 3253;

[2, 113, 11] 211311 211310 = 2 * 5 * 11 * 17 * 113;

[2, 241, 31] 224131 224130 = 2 * 3 * 5 * 31 * 241;

[7, 2, 39779] 7239779 7239778 = 2 * 7 * 13 * 39779;

[2, 6173, 53] 2617353 2617352 = 2^3 * 53 * 6173;

[131, 13, 11] 1311311 1311310 = 2 * 5 * 7 * 11 * 13 * 131;

[131, 13, 11] 1311311 1311310 = 2 * 5 * 7 * 11 * 13 * 131;
                       
[29, 5, 12721] 29512721 29512720 = 2^4 * 5 * 29 * 12721;

[131, 5, 5021] 13155021 13155020 = 2^2 * 5 * 131 * 5021;


Q2. Can the same be done with >3 distinct primes concatenated?

I found:
                   
 primes d   d-1   factorisation
 
[3, 13] 313 312 = 2^3 * 3 * 13;

[2, 2857] 22857 22856 = 2^3 * 2857;

[83, 167] 83167 83166 = 2 * 3 * 83 * 167;

[13, 2549] 132549 132548 = 2^2 * 13 * 2549;

[29, 16763] 2916763 2916762 = 2 * 3 * 29 * 16763;

 

***

Michael wrote:

 Q1: I also found the following:

a b c d d-1 factors of d-1
2 7 3253 273253 273252 2^2 x 3 x 7 x 3253
877 32063 13 8773206313 8773206312 2^3 x 3 x 13 x 877 x 32063
2 3 203539823 23203539823 23203539822 2 x 3 x 19 x 203539823

Q2: For 4 distinct primes I found:

a b c d e e-1 factors of e-1
3 2 7 19 32719 32718 2 x 3 x 7 x 19 x 41
2 73 3 13 273313 273312 2^5 x 3^2 x 13 x 73
7 2 131 23 7213123 7213122 2 x 3^2 x 7 x 19 x 23 x 131
3 61 13 23 3611323 3611322 2 x 3^2 x 11 x 13 x 23 x 61
251 2 5 11 2512511 2512510 2 x 5 x 7 x 11 x 13 x 251
521 5 2 11 5215211 5215210 2 x 5 x 7 x 11 x 13 x 521
3 2 13 28411 321328411 321328410 2 x 3 x 5 x 13 x 29 x 28411
3079 2 463 3 307924633 307924632 2^3 x 3^3 x 463 x 3079

And for 5 distinct primes I found:

a b c d e f f-1 factors of f-1
2 7 11 4133 3 271141333 271141332 2^2 x 3 x 7 x 11 x 71 x 4133
7 13 2 103 17 713210317 713210316 2^2 x 3 x 7 x 13 x 17 x 103 x 373
7 19 83 11 13 719831113 719831112 2^3 x 3 x 7 x 11 x 13 x 19^2 x 83
11 1061 2 3 61 1110612361 1110612360 2^3 x 3 x 5 x 11 x 13 x 61 x 1061

Using a different approach, I found these with 6 distinct primes:

k distinct primes factorization of k-1
2319103711 (2, 3, 19, 103, 7, 11) 2 x 3^4 x 5 x 7 x 11 x 19^2 x 103
101193759613 (101, 19, 37, 59, 61, 3) 2^2 x 3^2 x 11 x 19 x 37 x 59 x 61 x 101

***

Gennady wrote:

Q1.
273253 <=> 2^2 * 7 * 3 * 3253 = 273252
 
Q2.
 32719 <=> 3 * 2 * 7 * 19 * 41 = 32718
273313 <=> 2^5 * 73 *  3^2 * 13 = 273312
2512511 <=> 251 * 2 *  5 * 11 * 7 * 13 = 2512510
3611323 <=> 3^2 * 61 * 13 * 23 * 2 * 11 = 3611322
5215211 <=> 521 * 5 * 2 * 11 * 7 * 13 = 5215210
7213123 <=> 7 * 2 * 131 * 23 * 3^2 * 19 = 7213122
307924633 <=> 3079 * 2^3 * 463 * 3^3 = 307924632
321328411 <=> 3 * 2 * 13 * 28411 * 5 * 29 = 321328410

...

Q2.  5 prime divisors:
713210317 <=> 7 * 13 * 2^2 * 103 * 17 * 3 * 373 = 713210316
 
719831113 <=> 7 * 19^2 * 83 * 11 * 13 * 2^3 * 3 = 719831112
 
Exhausting search up to 10^9.

 

***

Giorgos wrote:

Here are some more examples for each category
The format of the answers will be 
d -> {a.b.c,...} -> Factorization of d-1

Q1.
273253 -> {2,7,3253} -> {{2, 2}, {3, 1}, {7, 1}, {3253, 1}}
1311311 -> {131,13,11} -> {{2, 1}, {5, 1}, {7, 1}, {11, 1}, {13, 1}, {131, 1}}
8773206313 -> {877,32063,13} -> {{2, 3}, {3, 1}, {13, 1}, {877, 1}, {32063, 1}}
90911909111 -> {90911,9091,11} -> {{2, 1}, {5, 1}, {11, 1}, {9091, 1}, {90911, 1}}

Q2.
4 primes
2512511 -> {251,2,5,11} -> {{2, 1}, {5, 1}, {7, 1}, {11, 1}, {13, 1}, {251, 1}}
7213123 -> {7,2,131,23} -> {{2, 1}, {3, 2}, {7, 1}, {19, 1}, {23, 1}, {131, 1}}
307924633 -> {3079,2,463,3} -> {{2, 3}, {3, 3}, {463, 1}, {3079, 1}}
1369559327391187 -> {13695593,2,739,1187} -> {{2, 1}, {3, 1}, {19, 1}, {739, 1}, {1187, 1}, {13695593, 1}}

5 primes
387731737123 -> {3877,3,173,71,23} -> {{2, 1}, {3, 1}, {23, 1}, {59, 1}, {71, 1}, {173, 1}, {3877, 1}}
16034313148127 -> {160343,13,1481,2,7} -> {{2, 1}, {7, 2}, {13, 1}, {53, 1}, {1481, 1}, {160343, 1}}

6 primes
2319103711 -> {2,3,19,103,7,11} -> {{2, 1}, {3, 4}, {5, 1}, {7, 1}, {11, 1}, {19, 2}, {103, 1}}
101193759613 -> {101,19,37,59,61,3} -> {{2, 2}, {3, 2}, {11, 1}, {19, 1}, {37, 1}, {59, 1}, {61, 1}, {101, 1}}
6563177017237 -> {6563,17,701,7,2,37} -> {{2, 2}, {3, 4}, {7, 1}, {17, 1}, {37, 1}, {701, 1}, {6563, 1}}
25457231353217 -> {25457,23,13,53,2,17} -> {{2, 7}, {13, 1}, {17, 1}, {23, 1}, {29, 1}, {53, 1}, {25457, 1}}
20903887167231097 -> {20903887,167,2,3,109,7} -> {{2, 3}, {3, 2}, {7, 1}, {109, 2}, {167, 1}, {20903887, 1}}

7 primes
58721341310113 -> {587,2,13,41,3,101,13} -> {{2, 5}, {3, 1}, {13, 2}, {41, 1}, {101, 1}, {587, 1}, {1489, 1}} (Note CR: prime 13 repeated, violates the rule)

***

 Adam wrote:

Period separates individual primes (so 2.3.11 stands for the example 2311) followed by the factorization of (d-1).

Length four:  3.2.7.19 and d-1=2*3*7*19*41
2.73.3.13 and d-1=2^5*3^2*13*73...........(only the primes in the concatenation appear in the factors of d-1)
251.2.5.11 and d-1=2*5*7*11*13*251
521.5.2.11 and d-1=2*5*7*11*13*521
7.2.131.23 and d-1=2*3^2*7*19*23*131
3.61.13.23 and d-1=2*3^2*11*13*23*61


Length five: 7.13.2.103.17 and d-1=2^2*3*7*13817*103*373

Length six:   2.3.19.103.7.11 and d-1=2*3^4*5*7*11*19^2*103

Considering these examples came so fast, I am somewhat surprised that a list of seven primes isn't appearing (yet).

 

***

Simon wrote:

Q1
2, 7, 3253, 273253, 273252, 2^2 * 3 * 7 * 3253
877, 32063, 13, 8773206313, 8773206312, 2^3 * 3 * 13 * 877 * 32063
90911, 9091, 11, 90911909111, 90911909110, 2 * 5 * 11 * 9091 * 90911


Q2
3, 2, 7, 19, 32719, 32718, 2 * 3 * 7 * 19 * 41
2, 73, 3, 13, 273313, 273312, 2^5 * 3^2 * 13 * 73
251, 2, 5, 11, 2512511, 2512510, 2 * 5 * 7 * 11 * 13 * 251
3, 61, 13, 23, 3611323, 3611322, 2 * 3^2 * 11 * 13 * 23 * 61
521, 5, 2, 11, 5215211, 5215210, 2 * 5 * 7 * 11 * 13 * 521
7, 2, 131, 23, 7213123, 7213122, 2 * 3^2 * 7 * 19 * 23 * 131
3079, 2, 463, 3, 307924633, 307924632, 2^3 * 3^3 * 463 * 3079
2, 149, 1013, 809, 21491013809, 21491013808, 2^4 * 11 * 149 * 809 * 1013

7, 13, 2, 103, 17, 713210317, 713210316, 2^2 * 3 * 7 * 13 * 17 * 103 * 373
7, 19, 83, 11, 13, 719831113, 719831112, 2^3 * 3 * 7 * 11 * 13 * 19^2 * 83
11, 1061, 2, 3, 61, 1110612361, 1110612360, 2^3 * 3 * 5 * 11 * 13 * 61 * 1061
5, 47, 3, 101, 1201, 54731011201, 54731011200, 2^7 * 3 * 5^2 * 47 * 101 * 1201

2, 3, 19, 103, 7, 11, 2319103711, 2319103710, 2 * 3^4 * 5 * 7 * 11 * 19^2 * 103
101, 19, 37, 59, 61, 3, 101193759613, 101193759612, 2^2 * 3^2 * 11 * 19 * 37 * 59 * 61 * 101

***

Emmanuel wrote:

Q1.
I only could find one more solution :
{a, b, c, d} = {2, 7, 3253, 273253}; d - 1 = 2^2*3*7*3253

Q2
I found several solutions with four primes :
 a          b          c        d           e                     factorization of  e - 1
2           73        3        13         273313          2^5*3^2*13*73
3           2          7        19         32719            2*3*7*19*41
3           61        13      23         3611323        2*3^2*11*13*23*61
7           2          131    23         7213123        2*3^2*7*19*23*131
251       2          5        11         2512511        2*5*7*11*13*251
521       5          2        11         5215211        2*5*7*11*13*521
If we admit repeated primes :
2           11        2        353       2112353        2^5*11*17*353
2           11        11      113       21111113      2^3*11^2*113*193

Here is a solution with five primes :
 a          b          c         d          e         f                         factorization of  f - 1
7           19        83       11        13      719831113        2^3* 3*7*11*13*19^2* 83
With repetition allowed :
2           2          71       43        3        2271433            2^3*3* 31* 43* 71
2           3          3         17        3        233173              2^2*3^3*17*127
3           3          2         3          17      332317              2^2*3^3* 17*181
3           61        13       2          3        3611323            2*3^2*11*13*23*61
7           13        2         103      17      713210317        2^2* 3*7*13*17*103*373
19         23        13       2          13      192313213        2^2* 3* 7* 13^2*19*23*31
47         7          3         13        3        4773133            2^2*3^2* 7*13*31*47
47         7          31       3          3        4773133            2^2*3^3* 7*13*31*47
53         2          3         3,         3        532333              2^2*3^4* 31* 53

Six primes :
a          b         c        d        e          f            g                         factorization of  g - 1
2          3        19      103    7          11        2319103711        2*3^4*5*7*11*19^2*103
101      19      37      59      61        3          101193759613    2^2*3^2*11*19*37*59*61*101
With repeated primes
2          2        3        2        7          3          223273                2^3*3^2*7*443
2          83      71      2        59        3          283712593          2^4*3*17*59*71*83
3          2        29      3        107      19        3229310719        2*3^2*17*19*29*107*179
5          2        2        23      3          41        52223341            2^2*3*5*13*23*41*71
11        2        3        11      11        11        1123111111           2*3*5*11^4*2557
13        3        5        3        2          101      133532101          2^2*3^2*5^2*13*101*113
19        2        3        13      2          13        192313213          2^2*3*7*13^2*19*23*31
19        2        31      3        2          13        192313213          2^2*3*7*13^2*19*23*31
97        2        2        19      3          7          97221937            2^4*3*7*19*97*157

Seven primes : I found no solutions with different primes, but :
a      b       c       d        e         f         g        h                       factorization of  h - 1
2      3      61      2        2        41      13      2361224113      2^4*3*13*17*41*61*89
3      2      5        11      23      3        71      3251123371      2*3^2*5*11*23*71*2011
7      2      2        2        2        19      3        72222193          2^4*3^4*7*19*419

Eight primes : I found no solutions with different primes, but :
2        2        2        3        3      11      11      3          2223311113          2^3*3^3*11^2*257*331
2        2        3        7        2      2        2        41        223722241            2^8*3*5*7^2*29*41
2        2        23      3        2      2        41      7          2223322417          2^4*3^2*7*23*41*2339
3        3        11      2        2      2        11      3          3311222113          2^5*3^3*11^2*19*1667
3        3        13      7        7      7        19      41        33137771941        2^2*3^2*5*7^3*13*19*41*53
3        7        19      7        7      3        2        7          371977327            2*3^3*7^4*19*151
5        2        3        13      3      7        11      11        52313371111        2*3^2*5*7*11^3*13*4799
7        2        2        3        2      2        7        3          72232273              2^4*3^2*7^2*29*353
7        2        2        3        7      2        11      3          722372113            2^4*3^2*7^2*11*41*227
7        37      3        29      3      3        2        7          7373293327          2*3^3*7^4*29*37*53
11      2        2        3        2      7        3        7          112232737            2^5*3^3*7^2*11*241
11      3        2        3        37      3      29      3          11323373293        2^2*3^6*7*11*29*37*47
11      3        3        29      3      37      2        3          11332933723        2*3^4*11*29*37*5927
31      2        3        5        3      3        3        41        3123533341          2^2*3^4*5*31*37*41^2
31      11      5        3        3      2        7        11        31115332711        2*3^4*5*7^2*11^3*19*31

I guess that there are infinitely many concatenations of  n  primes that give solutions when we allow repetitions.
But I've no idea if all primes must be different.

***

Oscar wrote:

Unlike puzzle 1220, I only considered concatenations involving no leading zeros.
Solutions below 10^13.

Two primes (rare!):
313   3 13
2916763   29 16763

Three primes:
1123   11 2 3
2113   2 11 3
2311   2 3 11
5741   5 7 41
224131   2 241 31
273253   2 7 3253
1311311   131 13 11
8773206313   877 32063 13
23203539823   2 3 203539823
90911909111   90911 9091 11
1312715766503   131 2 715766503

Four primes:
32719   3 2 7 19
273313   2 73 3 13
2512511   251 2 5 11
3611323   3 61 13 23
5215211   521 5 2 11
7213123   7 2 131 23
307924633   3079 2 463 3
321328411   3 2 13 28411
21491013809   2 149 1013 809

***

Paul Cleary sent his soluions to this puzzle a little bit later (Saturday 5, 2025, 3:06 am). This is why I couldn't publish his result together with the rest above, usually posted between 4-5 pm on each friday.

The worst this is that he already got two examples with seven distinct primes...!!! (in bold-red below)

 Here are his results:

{2,313,{3,13},312 -- 2^3 * 3 * 13}
{2,22857,{2,2857},22856 -- 2^3 * 2857}
{2,83167,{83,167},83166 -- 2 * 3 * 83 * 167}
{2,132549,{13,2549},132548 -- 2^2 * 13 * 2549}
{2,2916763,{29,16763},2916762 -- 2 * 3 * 29 * 16763}
{2,37100271,{37,100271},37100270 -- 2 * 5 * 37 * 100271}
{2,4331000231,{433,1000231},4331000230 -- 2 * 5 * 433 * 1000231}
{3,1123,{11,2,3},1122 -- 2 * 3 * 11 * 17}
{3,2113,{2,11,3},2112 -- 2^6 * 3 * 11}
{3,2311,{2,3,11},2310 -- 2 * 3 * 5 * 7 * 11}
{3,5741,{5,7,41},5740 -- 2^2 * 5 * 7 * 41}
{3,21373,{2,137,3},21372 -- 2^2 * 3 * 13 * 137}
{3,25641,{2,5,641},25640 -- 2^3 * 5 * 641}
{3,211311,{2,113,11},211310 -- 2 * 5 * 11 * 17 * 113}
{3,224131,{2,241,31},224130 -- 2 * 3 * 5 * 31 * 241}
{3,273253,{2,7,3253},273252 -- 2^2 * 3 * 7 * 3253}
{3,1311311,{131,13,11},1311310 -- 2 * 5 * 7 * 11 * 13 * 131}
{3,2617353,{2,6173,53},2617352 -- 2^3 * 53 * 6173}
{3,7239779,{7,2,39779},7239778 -- 2 * 7 * 13 * 39779}
{3,13155021,{131,5,5021},13155020 -- 2^2 * 5 * 131 * 5021}
{3,29512721,{29,5,12721},29512720 -- 2^4 * 5 * 29 * 12721}
{3,211436853,{2,11,436853},211436852 -- 2^2 * 11^2 * 436853}
{3,234893617,{2,3,4893617},234893616 -- 2^4 * 3 * 4893617}
{3,8773206313,{877,32063,13},8773206312 -- 2^3 * 3 * 13 * 877 * 32063}
{3,11909119091,{11909,11,9091},11909119090 -- 2 * 5 * 11 * 9091 * 11909}
{4,32719,{3,2,7,19},32718 -- 2 * 3 * 7 * 19 * 41}
{4,112531,{11,2,5,31},112530 -- 2 * 3 * 5 * 11^2 * 31}
{4,137229,{13,7,2,29},137228 -- 2^2 * 7 * 13^2 * 29}
{4,193117,{19,3,11,7},193116 -- 2^2 * 3 * 7 * 11^2 * 19}
{4,219337,{2,19,3,37},219336 -- 2^3 * 3 * 13 * 19 * 37}
{4,273313,{2,73,3,13},273312 -- 2^5 * 3^2 * 13 * 73}
{4,521731,{5,2,17,31},521730 -- 2 * 3^2 * 5 * 11 * 17 * 31}
{4,1713193,{17,13,19,3},1713192 -- 2^3 * 3 * 13 * 17^2 * 19}
{4,2512511,{251,2,5,11},2512510 -- 2 * 5 * 7 * 11 * 13 * 251}
{4,2712711,{271,2,7,11},2712710 -- 2 * 5 * 7 * 11 * 13 * 271}
{4,3611323,{3,61,13,23},3611322 -- 2 * 3^2 * 11 * 13 * 23 * 61}
{4,5215211,{521,5,2,11},5215210 -- 2 * 5 * 7 * 11 * 13 * 521}
{4,5715711,{571,5,7,11},5715710 -- 2 * 5 * 7 * 11 * 13 * 571}
{4,7213123,{7,2,131,23},7213122 -- 2 * 3^2 * 7 * 19 * 23 * 131}
{4,7517511,{751,7,5,11},7517510 -- 2 * 5 * 7 * 11 * 13 * 751}
{4,22713237,{2,2713,23,7},22713236 -- 2^2 * 7 * 13 * 23 * 2713}
{4,23661157,{2,3,661,157},23661156 -- 2^2 * 3 * 19 * 157 * 661}
{4,32225677,{3,2,22567,7},32225676 -- 2^2 * 3 * 7 * 17 * 22567}
{4,32271223,{3,2,271,223},32271222 -- 2 * 3 * 89 * 223 * 271}
{4,52720921,{5,2,7,20921},52720920 -- 2^3 * 3^2 * 5 * 7 * 20921}
{4,103941561,{1039,41,5,61},103941560 -- 2^3 * 5 * 41 * 61 * 1039}
{4,153161241,{1531,61,2,41},153161240 -- 2^3 * 5 * 41 * 61 * 1531}
{4,213320563,{2,13,3,20563},213320562 -- 2 * 3 * 7 * 13 * 19 * 20563}
{4,217166943,{2,17,1669,43},217166942 -- 2 * 17 * 43 * 89 * 1669}
{4,233083117,{2,3,308311,7},233083116 -- 2^2 * 3^3 * 7 * 308311}
{4,307924633,{3079,2,463,3},307924632 -- 2^3 * 3^3 * 463 * 3079}
{4,311103673,{3,11,103,673},311103672 -- 2^3 * 3 * 11 * 17 * 103 * 673}
{4,321328411,{3,2,13,28411},321328410 -- 2 * 3 * 5 * 13 * 29 * 28411}
{4,379337311,{379,337,3,11},379337310 -- 2 * 3^3 * 5 * 11 * 337 * 379}
{4,432997101,{43,2,997,101},432997100 -- 2^2 * 5^2 * 43 * 101 * 997}
{4,732272491,{73,227,2,491},732272490 -- 2 * 3^2 * 5 * 73 * 227 * 491}
{4,760859219,{7,60859,2,19},760859218 -- 2 * 7 * 19 * 47 * 60859}
{4,973229713,{97,3,2297,13},973229712 -- 2^4 * 3 * 7 * 13 * 97 * 2297}
{4,2699912113,{269,991,211,3},2699912112 -- 2^4 * 3 * 211 * 269 * 991}
{4,3543686761,{3,5,43,686761},3543686760 -- 2^3 * 3 * 5 * 43 * 686761}
{4,4121603311,{41,2,16033,11},4121603310 -- 2 * 3 * 5 * 11 * 19 * 41 * 16033}
{4,4314947919,{43,149,479,19},4314947918 -- 2 * 19 * 37 * 43 * 149 * 479}
{4,5276134711,{5,2,761347,11},5276134710 -- 2 * 3^2 * 5 * 7 * 11 * 761347}
{4,7293665359,{7,29,3,665359},7293665358 -- 2 * 3^3 * 7 * 29 * 665359}
{4,11554772131,{11,5,5477,2131},11554772130 -- 2 * 3^2 * 5 * 11 * 2131 * 5477}
{4,12619252381,{12619,2,5,2381},12619252380 -- 2^2 * 3 * 5 * 7 * 2381 * 12619}
{5,325711,{3,2,5,7,11},325710 -- 2 * 3^2 * 5 * 7 * 11 * 47}
{5,2675311,{2,67,5,3,11},2675310 -- 2 * 3 * 5 * 11^3 * 67}
{5,7235131,{7,2,3,5,131},7235130 -- 2 * 3 * 5 * 7 * 131 * 263}
{5,17325211,{17,3,2,5,211},17325210 -- 2 * 3 * 5 * 7 * 17 * 23 * 211}
{5,31124017,{3,11,2,401,7},31124016 -- 2^4 * 3^2 * 7^2 * 11 * 401}
{5,31157281,{3,11,5,7,281},31157280 -- 2^5 * 3^2 * 5 * 7 * 11 * 281}
{5,41321359,{41,3,2,13,59},41321358 -- 2 * 3^2 * 13 * 41 * 59 * 73}
{5,52413991,{5,2,41,3,991},52413990 -- 2 * 3 * 5 * 41 * 43 * 991}
{5,231760117,{2,3,17,6011,7},231760116 -- 2^2 * 3^4 * 7 * 17 * 6011}
{5,271141333,{2,7,11,4133,3},271141332 -- 2^2 * 3 * 7 * 11 * 71 * 4133}
{5,353711491,{3,5,37,11,491},353711490 -- 2 * 3 * 5 * 11 * 37 * 59 * 491}
{5,519231811,{5,19,2,3,1811},519231810 -- 2 * 3 * 5 * 19 * 503 * 1811}
{5,526389711,{5,2,6389,7,11},526389710 -- 2 * 5 * 7 * 11 * 107 * 6389}
{5,531371101,{5,3,13,71,101},531371100 -- 2^2 * 3 * 5^2 * 13 * 19 * 71 * 101}
{5,713210317,{7,13,2,103,17},713210316 -- 2^2 * 3 * 7 * 13 * 17 * 103 * 373}
{5,719831113,{7,19,83,11,13},719831112 -- 2^3 * 3 * 7 * 11 * 13 * 19^2 * 83}
{5,792532741,{79,2,5,3,2741},792532740 -- 2^2 * 3 * 5 * 61 * 79 * 2741}
{5,1110612361,{11,1061,2,3,61},1110612360 -- 2^3 * 3 * 5 * 11 * 13 * 61 * 1061}
{5,2313653443,{2,31,3,653,443},2313653442 -- 2 * 3 * 31 * 43 * 443 * 653}
{5,2321343311,{23,2,13,433,11},2321343310 -- 2 * 5 * 11 * 13 * 23 * 163 * 433}
{5,2775210371,{277,5,2,103,71},2775210370 -- 2 * 5 * 71 * 103 * 137 * 277}
{5,7432149929,{7,43,2,1499,29},7432149928 -- 2^3 * 7 * 29 * 43 * 71 * 1499}
{5,11224677311,{11,2,2467,7,311},11224677310 -- 2 * 5 * 7 * 11 * 19 * 311 * 2467}
{5,11741167131,{11,7,41,167,131},11741167130 -- 2 * 5 * 7 * 11 * 17 * 41 * 131 * 167}
{6,531319231,{5,3,13,19,2,31},531319230 -- 2 * 3^3 * 5 * 13 * 19 * 31 * 257}
{6,732134131,{7,3,2,13,41,31},732134130 -- 2 * 3 * 5 * 7 * 13 * 31 * 41 * 211}
{6,1132957211,{113,29,5,7,2,11},1132957210 -- 2 * 5 * 7 * 11 * 29 * 113 * 449}
{6,2319103711,{2,3,19,103,7,11},2319103710 -- 2 * 3^4 * 5 * 7 * 11 * 19^2 * 103}
{6,3791731273,{379,17,31,2,7,3},3791731272 -- 2^3 * 3 * 7 * 17 * 31 * 113 * 379}
{6,7732133137,{7,73,2,13,3,137},7732133136 -- 2^4 * 3^2 * 7 * 13 * 59 * 73 * 137}
{7,1732317511,{17,3,2,31,7,5,11},1732317510 -- 2 * 3 * 5 * 7 * 11 * 17 * 31 * 1423}
 {7,2547377911,{2,5,47,3,7,79,11},2547377910 -- 2 * 3^4 * 5 * 7 * 11^2 * 47 * 79}

Later Paul wrote this:

 "Thanks for your patience. It’s been a hectic weekend, and I apologize for the slow reply.
 After reviewing my work, I see that I misread the problem—d needed to be prime—and some of my submitted results aren’t. To preserve the integrity of your pages, please feel free to remove my submission. That error is entirely on me.Thanks for your understanding."

According with this all the submissions from Mr. Cleary to this puzzle and he next one, Puzzle 1228, must be forgotten.

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