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Problems & Puzzles:
Puzzles
Puzzle 1265 A
sort of “convolution” between a prime P1 and
its succesor 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
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n
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P1
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P2
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Produced Primes
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1
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3
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5
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53
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2
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2
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3
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5,23
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3
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31
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37
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347, 401, 3731
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4
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233
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239
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23539, 24133, 233239, 239233
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5
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211
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223
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2333, 2441, 21323, 22511, 223211
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6
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2861
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2879
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31489, 288979, 290761, 2863879, 2881861,
28792861
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7
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10711
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10723
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117833, 1081823, 10721723, 10733711,
107240711, 1071110723, 1072310711
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8
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99611
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99623
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1095733, 1095841, 10060723, 10061911,
99722611, 996209623, 996329611, 9962399611
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9
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513053
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513059
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51818359, 51818953, 513566059, 513572053,
5131043059, 5131103053, 51305813059,
513053513059, 513059513053
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10
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5486807
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5486813
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60354883, 60354937, 554168107, 5492293813,
5492299807, 54873556813, 548686186813,
548686786807, 54868075486813, 54868135486807
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11
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184700947
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184700959
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2031710537, 184885659947, 1847194170959,
1847194290947, 18470279400959,
184701131700959, 184701143700947,
1847009774700947, 18470094884700959,
184700947184700959, 184700959184700947
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Q1: Can you extend
the sequence beyond the eleventh term?
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