Problems & Puzzles: Puzzles

Puzzle 945. Modular generation of primes from a prime. JM Bergot sent the following nice puzzle: Example 1: Prime= 353=300+50+3 and 2=353 mod 3; 3=353 mod 50;  53=353 mod 300 with all 2, 3, 53 being primes. Example 2: Prime=1307=1000+300+0+7 (If there is a 0, ignore) 5=1307 mod 7; 107= 1307 mod 300; 307=1307 mod 1000 with all 5, 107 and 307 being primes. Q. Send your largest prime that produce primes for all the remainders according this procedure? (The fewer the 0's, the better).

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Contributions came from Vicente Felipe Izquierdo, Pierandrea Formusa and Giovanni Resta.

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Vicente wrote on March 11, 2019

Prime, #, Residues
17, 2, {7,3}
113, 3, {13,3,2}
1223, 4, {223,23,3,2}
12113, 5, {2113,113,13,3,2}
121283, 6, {21283,1283,283,83,3,2}
1214657, 7, {214657,14657,4657,2657,257,7,3}
12336353, 8, {2336353,336353,36353,6353,353,53,3,2}
121847393, 9, {21847393,1847393,847393,247393,7393,5393,293,83,2}
1215614657, 10, {215614657, 15614657, 5614657, 614657, 14657, 4657, 2657, 257, 7, 5}
12184279817, 11, {2184279817, 184279817, 84279817, 24279817, 279817, 79817, 9817, 7817, 617, 7, 3}

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Pierandrea wrote on March 14, 2019:

My largest prime (with zeros inside) is 10^2000+50053 and my largest prime (<50*10^9) without zeros inside is 49963422857.

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Giovanni wrote on March 14, 2019:

I've searched for the smallest zeroless prime
of length n, for n from 2 to 16. Here are the results:

  2 17
  3 113
  4 1223
  5 12113
  6 121283
  7 1214657
  8 12336353
  9 121847393
10 1215614657
11 12184279817
12 121681658963
13 1215613379633
14 12751425844733
15 121947642682373
16 1219213679413217

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Paul Cleary wrote on March 15, 2019:

Here is my largest prime with no zeros:- 726862627433

 

And with 1 zero :- 4231885021283

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