Puzzle 1265 A sort of “convolution” between a prime P1 and its succesor P2

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Problems & Puzzles: Puzzles

Puzzle 1265 A sort of “convolution” between a prime P1 and its successor P2

On March 17, 2026, Paolo Lava sent the followiing puzzle.

Let us consider a sort of “convolution” between a prime P1 and its succesor P2.

Here below an example to clarify the idea.

P1 = 2861; the next prime is P2 = 2879. Now, starting from P1 multiplied by 10^(number of digit of P2) we have:

28610000 + 2879 = 28612879 (not prime)
2861000 + 2879 = 2863879 (prime)
286100 + 2879 = 288979 (prime)
28610 + 2879 = 31489 (prime)
2861 + 2879 = 5740 (not prime)
2861 + 28790 = 31651 (not prime)
2861 + 287900 = 290761 (prime)
2861 + 2879000 = 2881861 (prime)
2861 + 28790000 = 28792861 (prime)

In total, we have 6 primes and prime 2861 together with its successor 2879 is the least prime to produce 6 different primes through this process.Note that it makes no difference whether you start this process with P1 or P2.

Again, we move between the concatenation of P1 and P2 (28612879) to end with the concatenation of P2 and P1 (28792861), or viceversa: we do not continue with 2861000…0002879 or 2879000…0002861 otherwise it is a never ending story.I searched for the least prime P1 that produces, together with its successor P2, n different primes.

I searched for the least prime P1 that produces, together with its successor P2, n different primes. Here are my resukts

n	P1	P2	Produced Primes
1	3	5	53
2	2	3	5,23
3	31	37	347, 401, 3731
4	233	239	23539, 24133, 233239, 239233
5	211	223	2333, 2441, 21323, 22511, 223211
6	2861	2879	31489, 288979, 290761, 2863879, 2881861, 28792861
7	10711	10723	117833, 1081823, 10721723, 10733711, 107240711, 1071110723, 1072310711
8	99611	99623	1095733, 1095841, 10060723, 10061911, 99722611, 996209623, 996329611, 9962399611
9	513053	513059	51818359, 51818953, 513566059, 513572053, 5131043059, 5131103053, 51305813059, 513053513059, 513059513053
10	5486807	5486813	60354883, 60354937, 554168107, 5492293813, 5492299807, 54873556813, 548686186813, 548686786807, 54868075486813, 54868135486807
11	184700947	184700959	2031710537, 184885659947, 1847194170959, 1847194290947, 18470279400959, 184701131700959, 184701143700947, 1847009774700947, 18470094884700959, 184700947184700959, 184700959184700947

Q1: Can you extend the sequence beyond the eleventh term?

From April 11 -17, April 2026, contributions came from J. M. Rebert, Jeff Heleen, Emmanuel Vantieghem, Gennady Gusev, Paul Cleary, Simon Cavegn, Oscar Volpatti.

***

J. M. Rebert wrote:

I found:

n: P1, P2, [n Produced primes]

12: 1002002467, 1002002473, [11022027143, 101202249767, 10021026672473, 10021026732467, 100201248702473, 10020025672002473, 10020025732002467, 100200247702002473, 100200248302002467, 1002002474002002467, 10020024671002002473, 10020024731002002467]

13: 1232268241, 1232268253, [13554950771, 124459093541, 1233500509253, 1233500521241, 12323914678253, 12323914798241, 123228056368253, 12322683762268241, 123226825332268253, 123226826532268241, 1232268242232268253, 12322682411232268253, 12322682531232268241]

***
Jeff wrote:

For n = 12, p1 = 1002002467, p2 = 1002002473:
10020024671002002473 is prime
1002002468002002473 is not prime
100200247702002473 is prime
10020025672002473 is prime
1002003469002473 is not prime
100201248702473 is prime
10021026672473 is prime
1003004469473 is not prime
101202249173 is not prime
11022027143 is prime
2004004940 is not prime
10020024731002002467 is prime
1002002474002002467 is prime
100200248302002467 is prime
10020025732002467 is prime
1002003475002467 is not prime
100201249302467 is not prime
10021026732467 is prime
1003004475467 is not prime
101202249767 is prime
11022027197 is not prime

For n = 13, p1 = 1232268241, p2 = 1232268253:
12322682411232268253 is prime
1232268242232268253 is prime
123226825332268253 is prime
12322683642268253 is not prime
1232269473268253 is not prime
123228056368253 is prime
12323914678253 is prime
1233500509253 is prime
124459092353 is not prime
13554950663 is not prime
2464536494 is not prime
12322682531232268241 is prime
1232268254232268241 is not prime
123226826532268241 is prime
12322683762268241 is prime
1232269485268241 is not prime
123228057568241 is not prime
12323914798241 is prime
1233500521241 is prime
124459093541 is prime
13554950771 is prime

I found no others < 10^9.

***
Emmanuel wrote:

P1 = 1002002467, P2 = 1002002473 producent 12 primes :
{11022027143,101202249767,10021026672473,10021026732467,100201248702473,
10020025672002473,10020025732002467,100200247702002473,100200248302002467,
1002002474002002467,10020024671002002473,10020024731002002467}

P1 = 1232268241, P2 = 1232268253 producent 13 primes :
{13554950771,124459093541,1233500509253,1233500521241,12323914678253,
12323914798241,123228056368253,12322683762268241,123226825332268253,
123226826532268241,1232268242232268253,12322682411232268253,12322682531232268241}

If P1, P2 would produce 14 primes then P1 > 10^11.

***
Gennady wrote:

12: 1002002467, 1002002473

Primes: [11022027143, 101202249767, 10021026672473, 10021026732467, 100201248702473, 10020025672002473, 10020025732002467, 100200247702002473, 100200248302002467, 1002002474002002467, 10020024671002002473, 10020024731002002467]

13: 1232268241, 1232268253

Primes: [13554950771, 124459093541, 1233500509253, 1233500521241, 12323914678253, 12323914798241, 123228056368253, 12322683762268241, 123226825332268253, 123226826532268241, 1232268242232268253, 12322682411232268253, 12322682531232268241]

***
Paul wrote:

I can confirm n=1 to 11 by Paolo Lava to be accurate, I was only able to get two more digits for n=12 and 13, here are the smallest solutions of each.

n = 12, prime pair = {1002002467,1002002473};

10 10020024671002002473 prime
9 1002002468002002473
8 100200247702002473 prime
7 10020025672002473 prime
6 1002003469002473
5 100201248702473 prime
4 10021026672473 prime
3 1003004469473
2 101202249173
1 11022027143 prime
0 2004004940
-1 11022027197
-2 101202249767 prime
-3 1003004475467
-4 10021026732467 prime
-5 100201249302467
-6 1002003475002467
-7 10020025732002467 prime
-8 100200248302002467 prime
-9 1002002474002002467 prime
-10 10020024731002002467 prime

n = 13, prime pair = {1232268241,1232268253};

10 12322682411232268253 prime
9 1232268242232268253 prime
8 123226825332268253 prime
7 12322683642268253
6 1232269473268253
5 123228056368253 prime
4 12323914678253 prime
3 1233500509253 prime
2 124459092353
1 13554950663
0 2464536494
-1 13554950771 prime
-2 124459093541 prime
-3 1233500521241 prime
-4 12323914798241 prime
-5 123228057568241
-6 1232269485268241
-7 12322683762268241 prime
-8 123226826532268241 prime
-9 1232268254232268241
-10 12322682531232268241 prime

I went up to the pair {103401850807, 103401850817} and no n = 14 or above was found, there were over 20 solutions with n = 13 up to this pair

***
Simon wrote:

12 1002002467 1002002473: 11022027143,101202249767,10021026672473,10021026732467,100201248702473,10020025672002473,10020025732002467,
100200247702002473,100200248302002467,1002002474002002467,10020024671002002473,10020024731002002467
13 1232268241 1232268253: 13554950771,124459093541,1233500509253,1233500521241,12323914678253,12323914798241,123228056368253,
12322683762268241,123226825332268253,123226826532268241,1232268242232268253,12322682411232268253,12322682531232268241
14 143042268079 143042268091: 1573464948989,14447269075991,14447269077179,143185310347091,143185310359079,1430565723058091,
143042411121268091,143042411133268079,14304226952142268079,143042268222042268091,143042268234042268079,1430422680933042268091,
14304226808043042268091,143042268079143042268091
15 844598501179 844598501239: 9290583513569,85304448625079,845443099740179,8446829610291239,8446829610891179,84460694716401239,
844599345777501239,844599345837501179,8445985856988501179,84459850962498501239,84459850968498501179,844598502083598501179,
8445985012634598501239,84459850118744598501239,844598501179844598501239
16 9899150975347 9899150975353: 108890660728823,999814248510053,999814248510647,9909050126328347,99001408904505347
989924996686275347,98991519652680975347,989915107433850975353,989915107434450975347,9899150985246150975353,
9899150985252150975347,98991509763429150975347,989915097544599150975353,989915097545199150975347,98991509753479899150975353,
98991509753539899150975347
17 18197101446031 18197101446037: 200168115906347,200168115906401,1837907246049137,1837907246049731,181989211561756037,181989211561816031,1819728341705146031,
181971032657471446031,1819710162800201446037,1819710162800801446031,18197101464228101446037,18197101464234101446031,181971014478507101446037,
181971014478567101446031,1819710144621297101446037,1819710144621897101446031,1819710144603718197101446031

Searched up to 37821000000000

***
Oscar wrote:

n = 12
p1 = 1002002467
p2 = 1002002473
S = {11022027143, 101202249767, 10021026672473, 10021026732467, 100201248702473, 10020025672002473, 10020025732002467, 100200247702002473, 100200248302002467, 1002002474002002467, 10020024671002002473, 10020024731002002467}

n = 13
p1 = 1232268241
p2 = 1232268253
S = {13554950771, 124459093541, 1233500509253, 1233500521241, 12323914678253, 12323914798241, 123228056368253, 12322683762268241, 123226825332268253, 123226826532268241, 1232268242232268253, 12322682411232268253, 12322682531232268241}

n = 14
p1 = 143042268079
p2 = 143042268091
S = {1573464948989, 14447269075991, 14447269077179, 143185310347091, 143185310359079, 1430565723058091, 143042411121268091, 143042411133268079, 14304226952142268079, 143042268222042268091, 143042268234042268079, 1430422680933042268091, 14304226808043042268091, 14304226807914304

***

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