Problems & Puzzles: Puzzles

Puzzle 925. Seven primes associated (Puzzle 348 revisited) In our puzzle 348, we asked to get examples larger and larger as the following one: Given three consecutive primes (p1, p2, p3) as (41, 43 47) the following four are primes too: a) p4 = p1&p2&p3 = 414347 b) p5 = p1+p2+p3 = 41+43+47 = 131 d) p6 = sum of trios of digits in p4 = 414+347 = 761 e) p7 = sum of digits in p4 = 4+1+4+3+4+7 = 23   Carlos Rivera contributed to the puzzle 348 with and example such that p1 is of 43 digits. Recently he got a larger example such that p1 is of 200 digits, p1=10^199+153480417.   Q1. Can you find larger examples?

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Contributions came from Jan van Delden and Emmanuel Vantieghem

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Jan wrote on May 18, 2018:

Still running on 336 digit primes (so the concatenation would be titanic), but for now: "26031913*10^196+20091996*10^98+811560859".

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One day after Jan wrote again:

Finally my computer said “beep”:
 

The numbers (p1,p2,p3) = 26031913*10^328+20091996*10^164+2448219000+(571,687,699)
are a solution where p1&p2&p3 is titanic. The larger primes (p1..p5) are certified by primo.

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Emmanuel wrote:

Here is my best result for puzzle 925 :

p1 = 5*10^262 + 64497
 

p2 = 5*10^262 + 64817

p3 = 5*10^262 + 64943

p4 = p1&p2&p3 (a bit too long to print out here)

p5 = 15*10^262 + 194257

p6 = 4363

p7 = 97

The primality of all these primes was proved by PRIMO.

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