Problems & Puzzles: Puzzles

Puzzle 995. Another sequence of primes Paolo Lava sent the following nice puzzle: Take a prime p(i) and concatenate it with p(i+1). If the result is prime then add p(i)+p(i+1) and concatenate the sum to p(i+2). If the result is still prime then take p(i)+p(i+1)+p(i+2) and concatenate it to p(i+3) and so on. Here the list of the minimum prime that holds this process for n step: n    p 1    2 2    31 3    331 4    832757 5    2683591 Example: Pi Sum & Pi+1 Primality 2683591 Start VERDADERO 2683609 26835912683609 VERDADERO 2683613 53672002683613 VERDADERO 2683621 80508132683621 VERDADERO 2683649 107344342683649 VERDADERO 2683657 134180832683657 VERDADERO Q. Which are the next terms for n = 6, 7, 8 ...?

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On weel 3-9 April, 2020, contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

I think the next set of consecutive primes is :

 

p(i)                    sum&p(i+k)
6363925717
6363925723     63639257176363925723
6363925727     127278514406363925727
6363925807     190917771676363925807
6363925829     254557029746363925829
6363925831     318196288036363925831
6363925853     381835546346363925853

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On April 13, 2020, both but independently, Jan van Delden and Giovanni Resta found the earliest set of seven consecutive primes:

p[i]                           sum&p[i+k]

 

1478441195963

1478441195981   14784411959631478441195981

1478441196079   29568823919441478441196079

1478441196083   44353235880231478441196083

1478441196091   59137647841061478441196091

1478441196107   73922059801971478441196107

1478441196121   88706471763041478441196121

1478441196131 103490883724251478441196131

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One day after Giovanni wrote this:

Soon after I wrote you with a solution for 7 prime I found
one with 8 primes starting at 8996779470869.

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